UNIVERSITY OF CALIFORNIA,
IRVINE
Electron Affinity in Approximate Density Functional Theory and
the Role of Semiclassics in Orbital-Free Potential-Density Functional Theory
DISSERTATION
submitted in partial satisfaction of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
in Chemistry
by
Dong Hyung Lee
Dissertation Committee:
Professor Kieron Burke, Chair
Professor Craig Martens
Professor Vladimir Mandelshtam
2010
c 2010 American Chemical Society
Chapter 3 c 2009 American Institute of Physics
Chapter 4 c 2010 Dong Hyung Lee
All other materials TABLE OF CONTENTS
Page
LIST OF FIGURES
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LIST OF TABLES
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ACKNOWLEDGMENTS
viii
CURRICULUM VITAE
ix
ABSTRACT OF THE DISSERTATION
xii
1 Introduction
2 Background
2.1 Hohenberg-Kohn theorem
2.2 Kohn-Sham equation . . .
2.3 Green’s function . . . . . .
2.4 Uniform semiclassics in 1D
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in Approximate DFT
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5 Semiclassical Orbital-Free Potential-Density Functional Theory
5.1 Uniform semiclassical density . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 The leading term . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Quantum corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Accuracy of Electron Affinities of Atoms
3.1 Introduction . . . . . . . . . . . . . . . .
3.2 Computational details . . . . . . . . . .
3.3 Results and discussion . . . . . . . . . .
3.4 Why do limited basis sets work? . . . . .
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4 Condition on the Kohn-Sham Kinetic Energy
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 Theory and Illustration . . . . . . . . . . . . . . . .
4.3 Large Z Methodology . . . . . . . . . . . . . . . . .
4.4 Results and Interpretation . . . . . . . . . . . . . .
4.5 Modern Parametrization of Thomas-Fermi Density
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . .
ii
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5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.1.3 The evanescent region . . . . . . . . . . . . . . . . .
Uniform kinetic energy density . . . . . . . . . . . . . . . . .
Different limit . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary of equations . . . . . . . . . . . . . . . . . . . . .
1D Applications . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Harmonic potential . . . . . . . . . . . . . . . . . . .
5.5.2 Morse potential . . . . . . . . . . . . . . . . . . . . .
Is this variational? . . . . . . . . . . . . . . . . . . . . . . .
Spherically symmetric 3D potential . . . . . . . . . . . . . .
5.7.1 Isotropic 3D harmonic potential . . . . . . . . . . . .
5.7.2 Bohr atom . . . . . . . . . . . . . . . . . . . . . . . .
Real atoms . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.1 Single-point calculation . . . . . . . . . . . . . . . . .
5.8.2 Semiclassical orbital-free potential-density functional
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6 Conclusion
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Bibliography
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Appendices
101
A
Construction of modified GEA . . . . . . . . . . . . . . . . . . . . . . . . . . 101
iii
LIST OF FIGURES
Page
3.1
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Comparison of KS potentials of Li− . The black line is essentially exact, using
a density from quantum Monte Carlo. The red (dashed) line is the LDA
potential on that density. The horizontal lines mark the HOMO (2s orbital)
energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of errors (∆) in ionization potentials and electron affinities in
first 2 rows of periodic table. Energies are evaluated with B3LYP density
functional(33; 34) evaluated on HF densities and on self-consistent densities
within AVDZ basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of errors (∆) in IP’s and EAs (in eV). Energies are evaluated with
PBE density functional evaluated on the AVDZ basis set, except the square
symbols, where the densities are found from a HF calculation. . . . . . . . .
Shifted exact vS (r) potential (eV). The HOMO from QMC is -0.62 eV, and the
HOMO from LDA/AV5Z is 0.80 eV. We shift the exact vS by the difference
between eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shifted exact vXC (r) potential (eV) in Li− . The HOMO from QMC is -0.62
eV, and the HOMO from LDA/AV5Z is 0.80 eV. We shift the exact vXC by
the difference between eigenvalues. . . . . . . . . . . . . . . . . . . . . . . .
Plot of the radial density errors for Li− with various approximations. SCF
densities are obtained with AVQZ. The exact density is from quantum Monte
Carlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Percentage error between c1 Z 2 + c2 Z 5/3 and ∆TS = TS − c0 Z 7/3 . . . . . . . .
Difference between TS /Z 7/3 and c0 + c1 Z −1/3 + c2 Z −2/3 as a function of Z −1/3 .
Percentage errors for atoms (from Z = 1 to Z = 92) using various approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Accurate numerical Φ(y) and parametrized Φ(y) can not be distinguished. .
Errors in the model, relative to numerical integrations. . . . . . . . . . . . .
Plot of the scaled radial densities of Ba and Ra using Eq.(4.33) and SCF
densities. TF scaled densities of Ba and Ra are on top of each other. . . . .
Plot of the scaled reduced density gradient s(r) (relative to the local Fermi
wavelength) vs. Z 1/3 r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plot of the scaled reduced Laplacian q(r) (relative to the local Fermi wavelength) vs. Z 1/3 r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
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4.9
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
Plot of the reduced density gradient t(r) (relative to the local screening length)
vs. Z 1/3 r. As r → ∞, the TF t → 0.7071. . . . . . . . . . . . . . . . . . . .
Contour of integration in the complex E-plane. . . . . . . . . . . . . . . . . .
1D systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of exact and semiclassical densities of a harmonic potential with
N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of exact and semiclassical KE densities of a harmonic potential
with N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of exact and semiclassical densities of Morse potential with N = 2.
Comparison of exact and semiclassical KE densities of Morse potential with
N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spherical 3D systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of exact and semiclassical densities of l = 0 orbitals with N = 3
for an isotropic 3D harmonic potential. Eq. (5.92) is used from r = 0 to rI . .
Comparison of exact and semiclassical densities of l = 1 orbitals with N = 3
for an isotropic 3D harmonic potential. Eq. (5.92) is used from r = 0 to rI . .
Comparison of exact and semiclassical densities of l = 2 orbitals with N = 3
for an isotropic 3D harmonic potential. Eq. (5.92) is used from r = 0 to rI . .
Comparison of exact and semiclassical densities of 1s1 2s1 3s1 4s1 . . . . . . . .
Comparison of exact and semiclassical densities of 2p1 3p1 4p1 . . . . . . . . . .
Comparison of exact and semiclassical densities of 3d1 4d1 . . . . . . . . . . .
Comparison of exact and semiclassical densities of 4f 1 . . . . . . . . . . . . .
Comparison of SCF-LDA and semiclassical densities of Kr. Eq. (5.93) is used
from r = 0 to rI . (l + 1/2)2 is used for all orbitals instead of l(l + 1). . . . . .
Comparison of SCF-LDA and semiclassical densities of Kr. Eq. (5.93) is used
from r = 0 to rI . (l + 1/2)2 is used for all orbitals instead of l(l + 1). . . . . .
SCF procedures for KS-DFT (left) and orbital-free potential-density functional theory (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of LDA-SCF and semiclassical SCF densities of Ar. . . . . . . .
Convergence of total energies in each iterations . . . . . . . . . . . . . . . .
v
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65
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92
92
LIST OF TABLES
Page
3.1
3.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
5.1
5.2
5.3
Errors in EAs (eV). Total energies for both neutral and negative atoms are
evaluated on HF-SCF densities with aug-cc-pVDZ basis set. . . . . . . . . .
Errors in EAs (eV). Total energies for both neutral and negative atoms are
calculated by SCF procedure with aug-cc-pVDZ basis set. . . . . . . . . . .
The coefficients in the asymptotic expansion of the KS kinetic energy and
various local and semilocal functionals. The fit was made to Z=24 (Cr), 25
(Mn), 30 (Zn), 31 (Ga), 61 (Pm), and 74 (W). The functionals of the last two
rows are defined in section 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . .
KS kinetic energy (T ) in hartrees and various approximations for alkali-earth
atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Non-interacting kinetic energy (in hartrees) for molecules, and errors in approximations. All values are evaluated on the converged KS orbitals and densities obtained with B88-PW91 functionals, and the MGEA4 kinetic energies
are evaluated using the TF and the GEA data from Ref. (57). . . . . . . .
Jellium surface kinetic energies (erg/cm2 ) and % error, which is (σSapp −
σSex )/σSex , of each approximation. . . . . . . . . . . . . . . . . . . . . . . . . .
104 × (γSeff (rS , N) − γSTF (rS , N)) in atomic units vs. N = Z for neutral jellium
spheres with rS = 3.93 with various functionals. As N = Z → ∞, γSeff tends
to the curvature kinetic energy of jellium, γS . . . . . . . . . . . . . . . . . . .
KS kinetic energy (T ) in hartrees and various approximations for noble atoms.
The values of βi are found by fitting Eq.(4.22) to the accurate numerical
solution, and those of αi are the parameters of small-y expansion (65). B is
given by 1.5880710226. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Various moments calculated with our model
j and with the models of Ref.
R
(p)
p Φ(x)
. . . . . . . . . . . . . . . .
(63; 64). Here Mj is given by dx x
x
The
√ percentage errors of densities and KE densities at the turning point (x =
− 2N) for a harmonic oscillator. . . . . . . . . . . . . . . . . . . . . . . . .
Errors of densities, kinetic energy, potential energy for a harmonic oscillator
(v(x) = x2 /2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Errors of kinetic energies and potential energies for a harmonic oscillator
(v(x) = x2 /2). The semiclassical density is re-normalized to N to calculate
kinetic energies from tDF (x) (Eq. (5.68)) and potential energies. . . . . . . .
vi
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21
34
36
38
39
40
42
45
48
70
71
71
5.4
5.5
Variational principle. The trial potential, e
v(x), is xp /2 and the exact (target)
potential is x4 /2. For all cases, the minimum of the total energy is at p = 4.
Variational principle for N = 1. The trial potential, e
v (x), is αx2 and the exact
2
(target) potential is 3x /4. The minimum of the total energy is at α = 3/4. .
vii
75
76
ACKNOWLEDGMENTS
I would like to thank my parents, parents-in-law, my two sisters and my brother-in-law for
all the support they have given me the past 5 or 6 years.
Especially, I really thank my wife, Yumi Kim, for the dedicated support.
I would also like to thank Prof. Kieron Burke for his great advise and should acknowledge the help of co-authors Dr. Lucian A. Constantin, Prof. John P. Perdew for the paper
“Condition on the Kohn-Sham kinetic energy, and modern parametrization of the ThomasFermi density” and Prof. Filipp U. Furche for the paper “Accuracy of Electron Affinities of
Atoms in Approximate Density Functional Theory”. I also want to thank all the members
in the Burke group, both past and present members and also everyone else whom I have
known at Rutgers university and UCI.
Finally I acknowledge funding from NSF (CHE-0809859) and the Korea Science and Engineering Foundation Grant (No. C00063) as well as copyright permissions from American
Institute of Physics (Reprinted with permission from Donghyung Lee, Lucian A. Constantin,
John P. Perdew, and Kieron Burke, J. Chem. Phys. 130, 034107 (2009). Copyright 2009,
American Institute of Physics.) for Chapter 4 and American Chemical Society (Reproduced
with permission from Donghyung Lee, Filipp Furche and Kieron Burke, J. Phys. Chem.
Lett., 2010, 1, 2124 (2010). Copyright 2010 American Chemical Society.) for Chapter 3.
viii
CURRICULUM VITAE
Dong Hyung Lee
EDUCATION
Doctor of Philosophy in Chemistry
University of California, Irvine
2010
Irvine, California
Master of Science in Chemistry
Seoul National University
2004
Seoul, Korea
Bachelor of Science in Chemistry
Seoul National University
2002
Seoul, Korea
TEACHING EXPERIENCE
Teaching Assistant
Teach discussion classes for general chemistry
Teaching Assistant
Teach general chemistry labs.
2008, Fall quarter
UCI
2005–2006
Rutgers Univ.
Teaching Assistant
Teach computational chemistry labs for general chemistry.
2002
Seoul Nat. Univ.
CONFERENCES AND PRESENTATIONS
• Controlling Diaza-Cope rearrangement reactions with resonance assisted hydrogen
bonds: Theoretical and experimental investigations
Donghyung Lee, Jik Chin, Fabrizio Mancin, Nirusha Thavarajah, Alan Lough, and
Doo Soo Chung in The 227th ACS National Meeting, Anaheim, CA, 2004
• Prediction of retention factors in micellar electrokinetic chromatography using a density functional theory and a semi-empirical method
Donghyung Lee, Sungu Hwang, and Doo Soo Chung in The 227th ACS National Meeting, Anaheim, CA, 2004
• Negative ions in density functional theory
Donghyung Lee and Kieron Burke, 46th Sanibel Symposium 26 February-3 March
2006, St. Simons Island, GA, USA
• Time dependent Density-Functional Theory: Prospects and Applications, 2nd International Workshop and School, Benasque, Spain, from August 27, 2006 to September
10, 2006
ix
• 2007 GORDON RESEARCH CONFERENCE on Time-Dependent Density-Functional
Theory
• Semiclassical Origin of Density Functionals
Attila Cangi, Peter Elliott, Donghyung Lee, and Kieron Burke in APS March meeting
2008, New Orleans, Louisiana
• Semiclassical Approaches in Density Functional Theory
Peter Elliott, Attila Cangi, Donghyung Lee, and Kieron Burke in APS March meeting
2008, New Orleans, Louisiana
• A new approach to density functional theory
Kieron Burke, Peter Elliott, Attila Cangi, and Donghyung Lee in APS March meeting
2008, New Orleans, Louisiana
• Exact condition on the non-interacting kinetic energy for real matter
Donghyung Lee and Kieron Burke in APS March meeting 2008, New Orleans, Louisiana
• Density Functional Theory and Semiclassical Methods
Peter Elliott, Donghyung Lee, Attila Cangi, and Kieron Burke in APS March meeting
2009, Pittsburgh, Pennsylvania
• Towards a semiclassical theory of electronic structure
Attila Cangi, Donghyung Lee, Peter Elliott, and Kieron Burke, in APS March meeting
2009, Pittsburgh, Pennsylvania
• Semiclassics in Density Functional Theory
Donghyung Lee, Attila Cangi, Peter Elliott, and Kieron Burke, in APS March meeting
2009, Pittsburgh, Pennsylvania
• Semiclassics in Orbital-free Potential-Density Functional Theory
Donghyung Lee, Attila Cangi, Peter Elliott, and Kieron Burke, in ACS, August 22-26,
2010, Boston, MA
PUBLICATIONS
1. Controlling Diaza-Cope Rearrangement Reactions with Resonance Assisted Hydrogen
Bonds J. Chin, F. Mancin, N. Thavarajah, D. Lee, A. Lough, and D. S. Chung, J. Am.
Chem. Soc. 125, 15276 (2003).
x
2. Abnormal Adsorption Behavior of Dimethyl Disulfide on Gold Surfaces J. Noh, S.
Jang, D. Lee, S. Shin, Y. J. Ko, E. Ito and S. Joo, Current Applied Physics, 7, 605-610
(2007).
3. Semiclassical Origins of Density Functionals P. Elliott, D. Lee, A. Cangi, and K. Burke,
Phys. Rev. Lett. 100, 256406 (2008).
4. Condition on the Kohn-Sham Kinetic Energy and Modern Parametrization of the
Thomas-Fermi Density D. Lee, L. A. Constantin, J. P. Perdew, K. Burke, J. Chem.
Phys. 130, 034107 (2009).
5. Leading Corrections to Local Approximations A. Cangi, D. Lee, P. Elliott, and K.
Burke, Phys. Rev. B 81, 235128 (2010).
6. Accuracy of Electron Affinities of Atoms in Approximate Density Functional Theory
D. Lee, F. Furche and K. Burke, J. Phys. Chem. Lett., 1, pp 2124-2129 (2010).
7. Finding Electron Affinities with Approximate Density Functionals D. Lee and K.
Burke, accepted to Mol. Phys (2010).
8. Semiclassical Approaches to One-Dimensional Potential and Coulomb Potential D. Lee,
P. Elliott, A. Cangi, and K. Burke, in preparation (2010).
xi
ABSTRACT OF THE DISSERTATION
Electron Affinity in Approximate Density Functional Theory and
the Role of Semiclassics in Orbital-Free Potential-Density Functional Theory
By
Dong Hyung Lee
Doctor of Philosophy in Chemistry
University of California, Irvine, 2010
Professor Kieron Burke, Chair
In this dissertation I discuss the effects of the self-interaction in atomic anions and the
role of semiclassics in the orbital-free (potential-)density functional theory. The first part
describes how accurate electron affinities can be obtained from the traditional basis-set
approaches, despite the positive highest occupied molecular orbital (HOMO) energy due
to the self-interaction. I also suggest an alternative way to calculate electron affinity using
approximate functionals evaluated on Hartree-Fock or exact-exchange densities for which
the extra electron is bound. In the second part, I study the asymptotic expansion of the
neutral-atom energy as the atomic number Z → ∞. The recovery of the correct asymptotic
expansion is an important condition on the Kohn-Sham kinetic energy for the accuracy of
approximate kinetic energy functionals for atoms, molecules, and solids. The density and
kinetic energy density as potential functionals are derived from the semiclassical Green’s
function. Using the density as a potential functional, I show the orbital-free potentialdensity functional calculation for atoms. I also present the energy potential functional in 1D
and spherical 3D, where the kinetic energy is obtained from the virial theorem.
xii
Chapter 1
Introduction
Density functional theory (DFT) has been successfully applied to many areas such as chemistry, solid-state physics, biology, and surface sciences(1). The basic theorems were proven
by Hohenberg and Kohn (HK)(2) in 1964, and the practical approach to real problems was
developed by Kohn and Sham(3). Kohn-Sham (KS) DFT is a rigorous way to deal with
real, interacting electron problems by mapping them into non-interacting electron problems,
which can be solved easily, and provides a balance between computational cost and accuracy.
KS-DFT enables us to handle much larger systems in which the traditional ab initio methods
can not. Although KS-DFT is formally rigorous, the exchange-correlation energy functional
(EXC [n]) needs to be approximated in practice. There are many approximations to EXC , such
as the local density approximation (LDA), generalized gradient approximation (GGA), hybrid functionals, and meta-GGAs. These approximate functionals provide suitable accuracy
for many chemistry and physics areas.
Despite many successful applications, there has been an argument about the applicability of
density functional approximations to atomic anions(4; 5). These approximate functionals can
not remove the self-repulsion of an electron in the classical Coulomb (or Hartree) potential.
1
This incomplete cancellation between the Hartree and exchange potentials is called the selfinteraction error (SIE). It is known that effects of the SIE decrease as the size of the system
increases, due to the spatial distribution of electron density. The SIE becomes drastic in small
atomic anions, making their highest occupied molecular orbital (HOMO) energies positive.
This result implies the HOMO is not a bound state, and casts doubt on the reliability of
DFT for electron affinity (EA) calculations.
In the original HK-DFT, the total energy is an explicit density functional and the interacting
kinetic energy, T [n], is also a density functional. However, in KS-DFT, the non-interacting
KS kinetic energy, TS [n], is evaluated on the KS orbitals and the kinetic correlation energy
(T [n] − TS [n]) is contained in the exchange-correlation energy functional. Due to this limitation, we need to solve the KS differential equation to obtain the density from the orbitals.
If we know TS as a density functional with sufficient accuracy, we can construct an orbitalfree density functional theory, yielding a computational method that scales linearly with the
number of particles. In particular, highly accurate TS [n] is necessary for orbital-free density
functional calculations, since TS [n] ≃ E[n] by the virial theorem. However, there are only a
small number of known conditions on TS (6) which approximate KS kinetic energy functional
should satisfy. For the EXC in KS-DFT, there are several exact conditions that EXC should
satisfy(7).
Despite successes achieved while focused on the density as a variable, the fundamental importance of the potential has emerged(8; 9). We recently published a paper about the
semiclassical origins of density functionals(10), in which we derive the semiclassical density
and kinetic energy density as potential functionals in 1D hard wall boundary systems. As
discussed in Ref. (9), if we obtain TS [v] and n[v] from the same semiclassical single-particle
Green’s function, we can find a variational equation for the total energy. Until now, we had
derived the semiclassical density and kinetic energy density for box boundary conditions.
However, in real systems, e.g., Coulomb potentials, there are always turning points where
2
the total energy is equal to the potential energy. Hence, it is necessary to study the uniform
semiclassical Green’s function for systems with turning points, in order to derive the density
and kinetic energy density as potential functionals.
In this dissertation, I will discuss the above problems. In Chapter 2, I will give a basic
background in DFT, the uniform semiclassical approximation, and the Green’s function
approach. In Chapter 3, I will discuss the effect of SIE in EA calculations with approximate
density functionals and explain how to get reasonable EAs from traditional approaches,
despite positive HOMO energies. In Chapter 4, I show that the asymptotic expansion of
total energies of neutral atoms with atomic number Z can be a very important condition for
high accuracy in non-interacting KS kinetic energies. In Chapter 5, I will discuss semiclassical
orbital-free potential-density functional theory using a Green’s function approach. Finally,
I conclude with an overview and discuss the limitations encountered in these methods and
their influence on electronic structure methods.
3
Chapter 2
Background
In this section, I will review the basics of DFT, Green’s function approaches, and uniform
semiclassical approximations in 1D.
2.1
Hohenberg-Kohn theorem
The first Hohenberg-Kohn (HK) theorem states that the ground state density determines
the external potential vext (r) uniquely up to an arbitrary constant. Consider the electron
density n(r) for the nondegenerate ground state of some N-electron system. It determines
N by simple quadrature, as well as vext (r) and the corresponding Hamiltonian. From this
Hamiltonian, we can determine the wavefunction and hence get all properties of the system.
′
Consider two external potentials, vext (r) and vext
(r), differing by more than a constant.
Assume each potential gives the same n(r) for its ground state. These two different potentials
provide two different Hamiltonians, Ĥ and Ĥ ′ , respectively, whose ground state densities are
the same although the normalized wavefunctions, Ψ and Ψ′ , are different. Taking Ψ′ as a
4
trial wavefunction of Ĥ, the variational principle yields
E0 < hΨ′ |Ĥ|Ψ′i
= hΨ′ |Ĥ ′ |Ψ′ i + hΨ′|Ĥ − Ĥ ′ |Ψ′ i
Z
′
′
= E0 + d3 r n(r) (vext (r) − vext
(r))
(2.1)
Similarly, taking Ψ as a trial wavefunction of Ĥ ′ ,
E0′ < hΨ|Ĥ ′ |Ψi
= hΨ|Ĥ|Ψi + hΨ|Ĥ ′ − Ĥ|Ψi
Z
′
= E0 + d3 r n(r) (vext
(r) − vext (r))
(2.2)
By adding Eqs. (2.1) and (2.2), we obtain a contradiction: E0 + E0′ < E0 + E0′ . This
result implies that n(r) can come from at most one external potential vext (r). Thus, n(r)
determines N and vext (r) and hence all properties of the ground state. In particular, the
ground state energy is a functional of the ground state electron density, and can be expressed
in terms of n0 (r):
E0 = T [n0 ] + Vee [n0 ] + Vext [n0 ]
Z
= T [n0 ] + Vee [n0 ] + d3 r n0 (r) vext (r) ,
(2.3)
where subscript ee indicates an electron interaction. The kinetic energy and electron-electron
interaction functionals are independent of the external potential, so the combination of these
5
two functionals is referred to as the universal functional or the Hohenberg-Kohn functional,
FHK .
E0 = FHK [n0 ] +
Z
d3 r n0 (r) vext (r)
(2.4)
FHK [n0 ] = T [n0 ] + Vee [n0 ]
(2.5)
An explicit form of either functional is not known. The electron-electron interaction energy
can be divided into the classical electrostatic energy (Coulomb energy, J[n](r)) and the
remaining part.
1
Vee [n0 ] =
2
Z
3
d r1
Z
d 3 r2
n(r1 )n(r2 )
+ UXC [n]
r12
= J[n] + UXC [n]
(2.6)
where UXC is the non-classical contribution, including all effects of self-interaction correction,
exchange and correlation. Since we do not know the explicit form of UXC , it needs to be
approximated.
The second HK theorem states that E[n], the functional that gives the ground state energy
of the system, delivers the lowest energy if and only if the input density is the true ground
state density, n0 (r). According to the variational principle,
E0 ≤ E[ñ] = T [ñ] + Vee [ñ] + Vext [ñ]
(2.7)
For any trial density ñ(r) that satisfies necessary boundary conditions (e.g., ñ(r) ≥ 0 with
R 3
d r ñ(r) = N), the energy E[ñ] is the upper bound to the true ground state energy E0 .
Equality in Eq. (2.7) results if and only if the input density is the true ground state electron
6
density. For any trial density ñ, there is a corresponding external potential ṽext (r), and hence
˜
a Hamiltonian Ĥ and a corresponding wavefunction Ψ̃. For the Hamiltonian Ĥ associated
with the true external potential vext (r),
E[ñ] = hΨ̃|Ĥ|Ψ̃i
= T [ñ] + Vee [ñ] +
E[n0 ] = hΨ0 |Ĥ|Ψ0 i
Z
d3 r ñ(r) vext (r)
E[ñ] ≥ E0 [n0 ]
2.2
(2.8)
Kohn-Sham equation
Although the basic theory for DFT was suggested by Hohenberg and Kohn, the kinetic and
exchange-correlation energy functionals are not known, and so cannot be used to solve real
problems. Kohn and Sham(3) suggested a way in which the Hohenberg-Kohn theorem could
be used in real calculations with acceptable accuracy. They map an interacting system into
a non-interacting system that has the same density as the interacting system. For a noninteracting, non-degenerate fermion ground state, exact wavefunctions are given by Slater
determinants. Thus, for the non-interacting system with a Hamiltonian including a local
effective potential veff (r):
N
N
1X 2 X
∇ +
veff (ri )
Ĥ = −
2 i=1 i
i=1
(2.9)
7
φ1 (r1 ) φ2 (r1 )
1 φ1 (r2 ) φ2 (r2 )
Ψs = √ ..
..
N! .
.
φ1 (rN ) φ2 (rN )
···
···
..
.
···
φN (r1 ) φN (r2 ) ..
.
φN (rN ) (2.10)
where the φi are the N lowest eigenstates of the one-electron Hamiltonian fˆKS . These orbitals
are determined by
fˆKS φi = ǫi φi ,
(2.11)
where fˆKS = − 21 ∇2 + veff (r) is the one-electron Kohn-Sham operator. Because the explicit
form of the true kinetic energy functional T [n] is not known, they suggested using Eq. (2.12),
the kinetic energy of a non-interacting system with the same density as the real, interacting
system.
N
TS [n] ≡ TS [Ψs ] = hΨs | −
1X 2
∇ |Ψs i
2 i=1 i
(2.12)
The difference between the non-interacting and true kinetic energies would then be treated
approximately. Kohn and Sham defined the total energy E[n] as
E[n] = TS [n] + Vext [n] + J[n] + EXC [n] ,
(2.13)
where EXC is the exchange-correlation energy, defined by
EXC [n] = T [n] − TS [n] + Eee [n] − J[n] .
(2.14)
This exchange-correlation energy functional includes the effects of exchange, correlation, and
the residual kinetic energy.
8
Using the Lagrange undetermined multiplier method with the constraint hφi |φj i = δij leads
the resulting equation:
Z
n(r2 )
1 2
3
+ vXC (r1 ) φi =
− ∇ + vext (r1 ) + d r2
2
r12
1 2
− ∇ + veff (r1 ) φi = ǫi φi
2
(2.15)
where vXC = δEXC /δn. According to Eq. (2.15), veff (r) depends on the density through
the Coulomb term and vXC [n]. Therefore, the KS one-electron Eq. (2.15) has to be solved
iteratively.
2.3
Green’s function
We derive an approximation to the many-particle density n(x) and the kinetic energy density
tS (x) for non-interacting, spinless fermions in a one-dimensional, smooth potential v(x). This
system is characterized by the solutions of the static Schrödinger equation:
2 dψ
p
+ [E − v(x)] ψ = 0 ,
dx2
(2.16)
where p is a constant depending on the units (i.e., in atomic units, p = 1/2) and E is the
energy eigenvalue.
Our derivation is based on the Green’s function g(x, x′ ; E), which, analogous to Eq. (2.16),
obeys
2 d g
p 2 + [E − v(x)] g = δ(x − x′ ) .
dx
(2.17)
9
The solutions of Eq. (2.17) can be expressed in terms of the independent solutions ψl (x)
and ψr (x) satisfying the boundary conditions on the left and on the right, respectively, i.e.,



 ψl (x)
ψr (x′ )
,
p W (E)
′
g(x, x ; E) =

′

 ψl (x )
ψr (x)
,
p W (E)
if x ≤ x′ ,
(2.18)
if x ≥ x′ .
where W (E) = ψl (x)∂x ψr (x) − ψr (x)∂x ψl (x) is the Wronskian and ∂x := ∂/∂x. From the
diagonal Green’s function g(x; E) := g(x, x′ = x; E) we extract the density via
1
n(x) =
2πi
I
C
dE g(x, E) ,
(2.19)
and the (non-interacting) kinetic energy density via
1
tS (x) =
2πi
I
C
dE [E − v(x)]g(x, E) ,
(2.20)
where E denotes a complex-valued energy and C any closed contour in the complex energyplane, enclosing all occupied poles on the real axis.
2.4
Uniform semiclassics in 1D
The Wentzel-Kramers-Brillouin (WKB) method provides approximate solutions to the 1D
Schrödinger equation
−
h̄2 d2 ψ(x)
+ v(x)ψ(x) = Eψ(x) ,
2m dx2
(2.21)
which can be rearranged into
d2 ψ(x) p2 (x)
+
ψ = 0,
dx2
h̄2
(2.22)
10
where p(x) =
p
2m(E − v(x)) is the semiclassical momentum. Using ψ(x) ∼ eif (x)/h̄ in Eq.
(2.22), expanding f (x) as a series of h̄, and collecting the first order of h̄ give us the WKB
solution:
ψ(x) ≃ p
R
i
C
e± h̄
p(x)
p(x)dx
.
(2.23)
However, this WKB solution diverges at any turning points present in the system, due to
p
the factor of 1/ p(x). Hence, uniform semiclassical approximations which give finite and
accurate descriptions at turning points are crucial.
There are two popular uniform semiclassical approximations(11; 12). Miller and Good(11)
developed a way to treat potentials with two turning points using a single, semiclassical
wavefunction without any patching scheme. However, this uniform approximation requires
us to map a given arbitrary potential into a harmonic oscillator well. Accordingly, the
original semiclassical momentum should also be mapped into the modified momentum of the
harmonic well. From Miller’s paper(12), the uniform approximate solutions to Schrödinger
equation for two-turning-point problems are given by
" 2/3 #
1/6
3
θ1 (x)
θ1 (x)
ψ1 (x) = 1/2 Ai −
p (x)
2
" 2/3 #
1/6
θ
(x)
3
ψ2 (x) = (−1)n 21/2 Ai −
θ2 (x)
,
p (x)
2
where p(x) =
p
θ1 (x) =
θ2 (x) =
(2.24)
2(E − v(x)) is the local wavenumber,
Z
x
dt p(t)
x1
Z x2
dt p(t) ,
(2.25)
x
11
(x1 , x2 ) are the left- and right-turning points respectively, and Ai is the Airy function.
According to the continuity of wavefunctions, the solutions should satisfy the following conditions at some point x0 :
ψ1 (x0 ) = ψ2 (x0 )
ψ1′ (x0 ) = ψ2′ (x0 ) .
(2.26)
Given this matching condition, the quantization condition becomes
1
Θ(E) = θ1 (x) + θ2 (x) = Nn +
π,
2
(2.27)
where the Nn is determined by
4 3/2
A
−
3π (n+1)/2
4 3/2
=
B
−
3π (n+2)/2
Nn =
1
,
2
1
,
2
n = 1, 3, 5, · · ·
n = 0, 2, 4, · · ·
(2.28)
The sets of {As } and {Bs } are the roots of the following equations:
Ai(−As ) = 0 ,
Ai(−Bs ) − 4Bs Ai′ (−Bs ) = 0 .
(2.29)
The numerical results of coefficients, Nn , can be found in Ref. (12).
12
Chapter 3
Accuracy of Electron Affinities of
Atoms in Approximate DFT
3.1
Introduction
Treating atomic anions with approximate density functional theory (DFT) has long been
controversial(4; 5). The net negative charge produces strong self-interaction errors (SIE)(13),
resulting in such a large upward bump in the effective potential that the last electron is
unbound.
In Fig. 3.1, we show the exact Kohn-Sham (KS) potential for Li− , found from the density
of a highly accurate quantum Monte Carlo (QMC) calculation with a zero-variance zerobias estimator(14), and by inversion of the KS equations. The HOMO is at ǫ2s = −A,
with A = 0.62 eV, the electron affinity (EA). We also show the KS potential when the
exchange-correlation (XC) contribution is evaluated using the local density approximation
Reproduced with permission from Donghyung Lee, Filipp Furche and Kieron Burke, J. Phys. Chem.
Lett., 2010, 1, 2124 (2010). Copyright 2010 American Chemical Society.
13
2
vs(r) (eV)
1
0
ε 2s
−
Li
ε 2s
QMC
LDA
-1
-2
0
2
4
6
8
10
12
r (a0)
Figure 3.1: Comparison of KS potentials of Li− . The black line is essentially exact, using
a density from quantum Monte Carlo. The red (dashed) line is the LDA potential on that
density. The horizontal lines mark the HOMO (2s orbital) energies.
(LDA) on this accurate density. The 2s orbital of this potential is a very sharp resonance,
at approximately 0.80 eV.
Formal theorists argue that approximate DFT does “not apply to negative atomic ions if
the orbital energy (is) not negative.” (15) Despite this, many have ignored these warnings,
calculated EAs within DFT, and found reasonable results using reasonable basis sets(4).
Because such calculations have positive HOMO energies for the anion, many authors report
EAs found in this way with a note of caution. More than a decade ago, these opposing views
were well-expressed in Refs (4) and (5), with the latter arguing for why such calculations
should be discounted on formal grounds, and the former demonstrating that no practical
problems arise, even with very large basis sets, and that useful results can be found for
many small molecules (16).
The fact remains that a formally problematic procedure yields physically meaningful results.
This strongly suggests that there is a systematic property to be explored. In the present
letter, we use DFT calculations with exact exchange to elucidate that structure and show
how the practical and formal are reconciled. Our analysis suggests a new practical solution
to the problem that is as accurate as any existing DFT method with fewer formal difficulties.
14
We begin with our notation and formalism. The KS equations for any atom or ion are
1 2
σ
− ∇ + vS (r) φiσ (r) = ǫiσ φiσ (r)
2
(3.1)
where vSσ (r) is a single, multiplicative spin-dependent KS potential and σ is a spin index (up
and down spins). The KS potential is written as a sum of three contributions:
σ
vSσ (r) = v(r) + vH [n](r) + vXC
[n↑ , n↓ ](r)
(3.2)
where v(r) = −Z/r for an atom, vH (r) is the Hartree potential, and the XC potential is
σ
vXC
[n↑ , n↓ ](r) =
δEXC [n↑ , n↓ ]
.
δnσ (r)
(3.3)
Thus, for either the exact or some approximate XC functional of the (spin)-densities, we
have a self-consistent set of equations.
Far from a nucleus, the Hartree potential decays as N/r, where N is the electron number.
The exact XC potential decays as
vXC (r) → −1/r,
r → ∞,
(3.4)
which is a pure exchange effect(17). For a neutral atom, Z = N, and vS (r) → −1/r exactly.
However almost all local and gradient-corrected functional approximations to vXC (r) decay
incorrectly with r, typically exponentially, as the density decays exponentially. This has only
a small effect on the density itself, but leads to very poor HOMO levels in such calculations
(errors of several eV). These are all manifestations of the infamous SIE.
A well-known cure for this problem in DFT is to use the optimized effective potential (OEP)
method(18), which finds the KS potential for an orbital-dependent functional. The simplest
15
case is to include exchange alone, often denoted EXX, producing a KS potential with the
correct asymptotic behavior and a HOMO energy that is close to −I for neutral systems.
A practical and accurate approximation to OEP, called localized Hartree-Fock (LHF), was
developed by Görling and co-workers (19; 20). The total energies of EXX are practically
indistinguishable from those of a standard Hartree-Fock (HF) calculation (21).
While awkward and embarrassing for DFT enthusiasts, this problem is not fatal for neutral
atoms and cations, because one extracts the total energy by applying the energy functional
to the self-consistent density, and need never look at the orbital energies. Such total energies
(and especially energy differences) have been found to have chemically useful accuracy(22).
But for anions, the problem does appear fatal, since the exponential decay of the approximate
vXC (r) leaves a KS potential that behaves as
vS (r) → +1/r,
r → ∞,
(3.5)
for an anion using an approximate XC functional. This produces a large positive bump in
the potential, especially for small systems, and usually raises the HOMO above the zero of
the potential (value at ∞). As we show in Fig. 3.1, the LDA HOMO energy is 0.83 eV, and
the last orbital is not an eigenstate, but a very sharp resonance. The KS potential decays
slowly and the classical outer turning point is at 17 Å. The true self-consistent ground state,
using an approximate functional, is found when a sufficiently large fraction of an electron
has escaped from the system (tunnelled to ∞), reducing the HOMO energy to zero. We
avoid this sorry fate in Fig. 3.1 by evaluating the LDA approximation to the KS potential
on the exact density, not a self-consistent density. Note that hybrid functionals mix about
25% of exact exchange with a GGA. This reduces the barrier by about 25%, but does not
rectify the problem.
Long ago, Shore et al(13) studied the local density approximation for the H− ground state.
16
They confined electrons in a cavity by adding a spherical hard wall and calculated HOMO
and total energies, varying the position of the wall (RB ). They found that there was a
plateau-like region of the energies in 15 a.u. < RB < 30 a.u. To obtain the asymptotic
solution of H− as RB → ∞, they varied the number of core electrons at r < 25 a.u. and their
wavefunction to determine the energy minimum. The energy minimum occurred when 1.7
electrons were localized near the nucleus, 0.3 electrons were delocalized in the asymptotic tail
of the density, and the HOMO eigenvalue became zero. At about the same time, Schwarz(23)
noted that both O− and F− are unstable using Xα(24) (a precursor to LDA(3)) because
HOMO eigenvalues are positive.
Galbraith and Schaefer(4) claim the applicability of approximate DFT for negative ions. If
basis sets such as Dunning’s augmented correlation consistent polarized valence double zeta
(aug-cc-pVDZ) basis set(25; 26) or larger are used, DFT methods with GGA and hybrid
functionals can be applied to negative ions such as F− , whose outermost electron has a
positive HOMO. Subsequently Rösch and Trickey (5) correctly point out that exact DFT
itself has no difficulty for anions, but that the problem is with approximate functionals, as
is clearly illustrated in Fig. 3.1. They claim that there are no difficulties with the physical
significance of orbital eigenvalues, positive values are permissible, and that problems can be
masked by finite basis sets. Although correct, none of this necessarily implies that accurate
numbers cannot be extracted from finite basis sets. Later, careful calculations by Jarecki
and Davidson(15) showed that, for F− , there are two plateau regions of total energy as a
function of basis-set size. On the first plateau, both total energy and HOMO appear to
converge, the latter to a positive value. Beyond that, the basis set probes the outside of the
barrier, some density leaks out, and the HOMO falls to zero. But the density change is so
small that the total energy barely changes. Eventually, the electron completely escapes and
the HOMO falls even further.
A naive solution to this difficulty is to perform HF or the DFT version, LHF, for such systems,
17
which produce negative HOMOs for the anions. However, for small anions, correlation effects
are large, so HF total energy differences are highly inaccurate. In fact, from the total energies,
many of the anions are unbound. One can also use Koopmans’ theorem, where I is estimated
as −ǫHOMO . While an improvement, the mean absolute errors (0.5 eV) are much larger
than DFT methods including correlation with a basis set. An alternative solution to the
SIE problem is to perform self-interaction corrected local spin density approximation (SICLSDA)(27). In Ref. (28), Cole and Perdew applied SIC-LSDA to calculate EAs for atomic
systems, Z < 86, significantly reducing errors relative to LSDA, but much less accurate than
results produced here with GGAs, hybrids, and meta-GGAs.
3.2
Computational details
In all our calculations, the total energies of neutral atoms and ions are calculated using
the usual self-consistent unrestricted HF, LHF, and KS-DFT. The approximate functionals in DFT calculations are LDA (S-VWN5)(29; 30; 31), PBE(32), hybrid (B3LYP(33; 34)
and PBE0(35)), and meta-GGA (TPSS(36)) functionals. We use Dunning’s augmented
correlation-consistent pVXZ (aug-cc-pVXZ, X = D, T, Q, and 5, AVXZ in this paper) basis
sets(25; 26). For the LHF calculations for anions, we calculate the Slater potential numerically everywhere to get accurate results. The calculations with basis sets are performed with
TURBOMOLE 6.2 (37). For the special cases of H− and Li− we perform fully numerical
DFT calculations using an OEP code(38) to calculate vS (r), vXC (r), and the densities using
EXX. Since this code makes a spherical approximation, we do not use it for non-spherical
cases. To calculate approximate functionals on HF densities, we perform unrestricted HF
calculations on both neutral and negative atoms. Then, we evaluate the total energies of
atoms using HF orbitals, so the kinetic energies are those of HF.
18
Table 3.1: Errors in EAs (eV). Total energies for both neutral and negative atoms are
evaluated on HF-SCF densities with aug-cc-pVDZ basis set.
H
Li
B
C
O
F
MAE1
Na
Al
Si
P
S
Cl
MAE
total
1
3.3
EA
Exp
0.75
0.62
0.28
1.26
1.46
3.40
0.55
0.43
1.39
0.75
2.08
3.61
LDA
0.13
-0.04
0.36
0.45
0.41
0.59
0.33
0.04
0.17
0.21
0.13
0.27
0.36
0.20
0.26
PBE
-0.11
-0.12
0.26
0.23
0.14
0.12
0.16
-0.01
0.13
0.11
0.03
0.06
0.08
0.07
0.12
∆EA
B3LYP
0.03
-0.14
0.04
-0.04
0.01
-0.04
0.05
-0.05
-0.08
-0.13
0.02
0.01
0.01
0.05
0.05
PBE0
-0.17
-0.14
0.17
0.12
-0.14
-0.18
0.15
-0.05
0.10
0.09
-0.03
0.00
0.03
0.05
0.10
TPSS
0.01
-0.05
0.12
0.12
-0.09
-0.07
0.08
0.03
0.06
0.07
0.03
0.01
0.02
0.04
0.06
Mean absolute error
Results and discussion
Our suggestion is to perform calculations that include exact exchange (either LHF or HF) for
self-consistent densities, but evaluate an approximate functional on that density to obtain
the energy. Such a procedure has its own drawbacks, but avoids all the pitfalls mentioned
above.
In Table 3.1, we calculate EAs using HF densities for both neutral and negative atoms.
The B3LYP results were used in Fig. 3.2. Our EAs are much more accurate than the
corresponding ionization potentials (typically by about a factor of 2), with mean average
errors below 0.1 eV. These results change little when the basis set is expanded (either larger
valence space or more diffuse functions), and our new method, using HF densities, has a
19
0.5
IP
EA on HF densities
EA (B3LYP/AVDZ)
0.4
∆ (eV)
0.3
0.2
0.1
0
-0.1
-0.2
0
2
4
6
8
Z
10
12
14
16
18
Figure 3.2: Comparison of errors (∆) in ionization potentials and electron affinities in first
2 rows of periodic table. Energies are evaluated with B3LYP density functional(33; 34)
evaluated on HF densities and on self-consistent densities within AVDZ basis set.
well-defined basis-set limit. The traditional method of limited basis sets (LBS) can only
work until the basis set probes the decay of the positive barriers of the type shown in Fig.
3.1. Similarly good results are found for the PBE generalized gradient approximation(32)
and its hybrid, PBE0(35). The best results are with the TPSS (36) meta-GGA with HF
densities. The results for all functionals are better than those of ionization potentials for
these elements. They also follow the usual trends for approximate functionals. The more
sophisticated functionals reduce errors by a factor of 2-5 relative to LDA.
In Fig. 3.3, we repeat Fig. 3.2, but now for the PBE functional. Comparison of the two
shows the more systematic (and often larger) errors of the non-empirical GGA versus the
empirical hybrid B3LYP. Interestingly, with our HF method for neutral and negative atoms,
PBE does as well as PBE0. B3LYP and the meta-GGA, TPSS, give the best total MAE.
We find almost identical results if we use LHF instead of HF densities.
20
0.5
IP
EA on HF densities
EA (PBE/AVDZ)
0.4
∆ (eV)
0.3
0.2
0.1
0
-0.1
-0.2
0
2
4
6
8
Z
10
12
14
16
18
Figure 3.3: Comparison of errors (∆) in IP’s and EAs (in eV). Energies are evaluated with
PBE density functional evaluated on the AVDZ basis set, except the square symbols, where
the densities are found from a HF calculation.
Table 3.2: Errors in EAs (eV). Total energies for both neutral and negative atoms are
calculated by SCF procedure with aug-cc-pVDZ basis set.
H
Li
B
C
O
F
MAE
Na
Al
Si
P
S
Cl
MAE
total
EA
Exp
0.75
0.62
0.28
1.26
1.46
3.40
0.55
0.43
1.39
0.75
2.08
3.61
-
LDA
0.15
-0.03
0.44
0.53
0.58
0.74
0.41
0.07
0.21
0.20
0.25
0.31
0.35
0.23
0.32
PBE
-0.07
-0.11
0.32
0.31
0.30
0.28
0.23
0.01
0.15
0.11
0.09
0.10
0.09
0.09
0.16
21
∆EA
B3LYP
0.06
-0.12
0.10
0.03
0.12
0.06
0.08
-0.02
-0.04
-0.12
0.09
0.04
0.02
0.06
0.07
PBE0
-0.16
-0.13
0.19
0.15
-0.07
-0.11
0.13
-0.03
0.11
0.08
0.00
0.01
0.02
0.04
0.09
TPSS
0.02
-0.04
0.16
0.18
0.03
0.05
0.08
0.04
0.09
0.07
0.07
0.03
0.02
0.05
0.07
2
vs(r) (eV)
1
Li
−
0
QMC
LDA on QMC
shifted QMC
-1
-2
0
5
10
15
20
25
30
35
r (a0)
Figure 3.4: Shifted exact vS (r) potential (eV). The HOMO from QMC is -0.62 eV, and
the HOMO from LDA/AV5Z is 0.80 eV. We shift the exact vS by the difference between
eigenvalues.
3.4
Why do limited basis sets work?
If the last electron is unbound in a pure DFT calculation, one can reasonably ask why
limited-basis calculations yield sensible answers at all. To confirm that they do work, in
Table 3.2, we report the results of using a limited basis set. The B3LYP and PBE numbers
were used in Figs. 3.2 and 3.3, respectively. Without addressing the formal issues, these
results are entirely sensible and very close to those of Table 3.1. MAE’s are almost the same
(slightly worsened), and individual differences are almost all within the MAE of the given
functional. How can such sensible results and excellent agreement with Table 3.1 come from
such an apparently ill-defined procedure?
The original EXC defined by Kohn and Sham(3) was for fixed particle number. This means
the KS potential is undefined up to an arbitrary constant. The density is unaffected by an
arbitrary shift of the potential. Thus positive orbital energies per se do not mean a density
or total energy is inaccurate. However, in real calculations with approximate XC functionals,
we conventionally set vS (r → ∞) to zero. Thus, the electron is unbound if its orbital energy
is positive, as in Fig. 3.1. Consider a model in which we add to the potential Cθ(Rc − r),
22
2
vXC(r) (eV)
0
-2
Li
-4
−
-6
QMC
LDA on QMC
shifted QMC
-8
-10
0
5
10
15
20
25
30
35
r (a0)
Figure 3.5: Shifted exact vXC (r) potential (eV) in Li− . The HOMO from QMC is -0.62 eV,
and the HOMO from LDA/AV5Z is 0.80 eV. We shift the exact vXC by the difference between
eigenvalues.
where θ(r) is the Heaviside step function and Rc is a very large fixed distance. As long
as our basis sets do not stretch out to Rc , the anion will appear perfectly stable and have
a well-defined limit for its density. If C is large enough, the HOMO will be positive. In
Fig. 3.4, we have performed just such a procedure for Li− , with C = 1.42 eV, and choosing
Rc = 1000Å, arbitrarily. This slightly inaccurate KS potential yields essentially the exact
anionic density, produces as accurate an anionic energy as the approximate functional used
to evaluate it, but has a positive HOMO of 0.80 eV.
Now then the question becomes: Do approximate functionals, complete with SIE, really
accurately mimic such a shifted KS potential? The answer has long been known to be
yes(39), and we show this in Fig. 3.5. Here we include only the XC portion, in order to
zoom in on the region where the exact and approximate potentials differ. The LDA potential
almost exactly follows the shifted exact potential, once the outer shell (2s) is reached. Does
it converge to an accurate energy and density? The answer is generally yes, if the functional
is accurate except for producing the wrong asymptote.
In Fig. 3.6, we show that the errors in density for the Li− anion are only a few percent, and
23
0.02
Li
2
4πr ∆n(r)
0.01
−
0
-0.01
HF
LHF
PBE
B3LYP
-0.02
-0.03
-0.04
0
5
10
15
20
25
30
r (a0)
Figure 3.6: Plot of the radial density errors for Li− with various approximations. SCF
densities are obtained with AVQZ. The exact density is from quantum Monte Carlo.
are comparable for all methods.
Finally, we investigate the dependence of HOMOs and total energies on basis sets by adding
more diffuse functions for F− . Moving from singly augmented to quadruply augmented
AVTZ basis functions, the HOMO drops from 1.57 eV down to 0.75 eV, but the total energy
of the anion changes by only 2 mH. We also checked the use of a logarithmic (diffusive) grid,
but this had effects only in the 50 µH regime.
In summary, we have suggested an alternative method for calculating EAs of small anions
that resolves the dilemma of positive orbital energies and has a well-defined basis set limit.
By evaluating the energies on LHF or HF densities, in which the last electron is properly
bound with a negative HOMO, accurate and sensible results are obtained. We have also
shown that the consistency and accuracy of using limited basis sets for SCF calculations
with approximate functionals can be understood, despite the positive HOMO energies of the
anions. But an advantage of our method is that the basis-set limit is always well-defined.
Using limited basis sets could run into difficulty if the self-interaction barrier becomes too
narrow or insufficiently high, making it impossible to find a plateau region. Of course, by
evaluating the potential with one functional while evaluating the energy with another, various
well-known complications arise, such as in the calculation of forces. But these difficulties are
24
far less subtle and challenging than those of positive HOMOs.
Finally, we note that even our method will fail if the (L)HF density is insufficiently accurate or
the approximate functional does not provide accurate energies. Thus we expect comparable
accuracy to that found here for molecular valence anions, but the weakly bound states such
as dipole- or higher multipole-bound anions(40) will be much more challenging and may
require self-interaction-free energy functionals along with a correct treatment of dispersion.
25
Chapter 4
Condition on the Kohn-Sham Kinetic
Energy, and Modern Parametrization
of the Thomas-Fermi Density
4.1
Introduction
Ground-state Kohn-Sham (KS) density functional theory (DFT) is a widely-used tool for
electronic structure calculations of atoms, molecules, and solids (1), in which only the density
functional for the exchange-correlation energy, EXC [n], must be approximated. But a direct,
orbital-free density functional theory could be constructed if only the non-interacting kinetic
energy, TS , were known sufficiently accurately as an explicit functional of the density (41).
Using it would lead automatically to an electronic structure method that scales linearly with
the number of electrons N (with the possible exception of the evaluation of the Hartree
energy). Thus the KS kinetic energy functional is something of a holy grail of density
Reprinted with permission from Donghyung Lee, Lucian A. Constantin, John P. Perdew, and Kieron
Burke, J. Chem. Phys. 130, 034107 (2009). Copyright 2009, American Institute of Physics.
26
functional purists, and interest in it has recently revived (42).
In this work, we exploit the “unreasonable accuracy” of asymptotic expansions (43; 44),
in this case for large neutral atoms, to show that there is a very simple condition that
approximations to TS must satisfy, if they are to attain high accuracy for total energies
of matter. By matter, we mean all atoms, molecules, and solids that consist of electrons
in the field of nuclei, attracted by a Coulomb potential. The condition is to recover the
(known) asymptotic expansion of TS /Z 7/3 for neutral atoms, in powers of Z −1/3 . By careful
extrapolation from accurate numerical calculations up to Z ∼ 90, we calculate the coefficients
of this expansion. We find that the usual gradient expansion, derived from the slowlyvarying gas, but applied to essentially exact densities, yields only a good approximation to
these coefficients. Thus, all new approximations should either build in these coefficients, or
be tested to see how well they approximate them. We perform several tests, using atoms,
molecules, jellium surfaces, and jellium spheres, and analyze two existing approximations. In
Ref. (45), a related method was used to derive the gradient coefficient in modern generalized
gradient approximations (GGA’s) for exchange. Given this importance of N = Z → ∞ as
a condition on functionals, we revisited and improved upon the existing parametrizations of
the neutral-atom Thomas-Fermi (TF) density. The second-half of the paper is devoted to
testing its accuracy.
4.2
Theory and Illustration
For an N-electron system, the Hamiltonian is
Ĥ = T̂ + V̂ext + V̂ee ,
(4.1)
27
where T̂ is the kinetic energy operator, V̂ext the external potential, and V̂ee the electronR
electron interaction, respectively. The electron density n(r) yields N = d3 r n(r), where N
is the particle number.
To explain asymptotic exactness, we (re)-introduce the ζ-scaled potential (46) (which is
further discussed in Ref. (10)), given by
ζ
vext
(r) = ζ 4/3 vext (ζ 1/3 r),
N → ζN,
(4.2)
ζ
where vext (r) is the external potential, and the Thomas-Fermi expectation value is Vext
[n] =
ζ 7/3 Vext [n]. In this ζ-scaling scheme, nuclear positions Rα and charges Zα of molecules are
scaled into ζ −1/3 Rα and ζZα respectively. In a uniform electric field, E → ζ 5/3 E. For neutral
atoms, scaling ζ is the same as scaling Z, producing an asymptotic expansion for the total
energy of neutral atoms (43; 47; 48; 49; 50),
E = −c0 Z 7/3 − c1 Z 2 − c2 Z 5/3 + · · · ,
(4.3)
where c0 = 0.768745, c1 = −1/2, c2 = 0.269900, and Z is the atomic number. This large Zexpansion gives a remarkably good approximation to the Hartree-Fock energy of the neutral
atoms, with less than a 10% error for H, and less than 0.5% error for Ne. By the virial
theorem for neutral atoms, T = −E, and T ≃ TS to this order in the expansion (since
the correlation energy is roughly ∼ Z). Hence, the non-interacting kinetic energy has the
following asymptotic expansion.
TS = c0 Z 7/3 + c1 Z 2 + c2 Z 5/3 + · · ·
(4.4)
We say that an approximation to the kinetic energy functional is asymptotically exact to
the p-th degree if it can reproduce the exact c0 , c1 , . . . , cp . The three displayed terms in Eq.
28
(4.3) constitute the second-order asymptotic expansion for the total energy of neutral atoms,
and we expect that this asymptotic expansion is a better starting point for constructing a
more accurate approximation to the kinetic energy functional than the traditional gradient
expansion approximation (GEA).
The leading term in Eq. (4.4) is given exactly by a local approximation to TS (TF theory),
but the leading correction is due to higher-order quantum effects, and only approximately
given by the gradient expansion evaluated on the exact density. However, these coefficients
are vital to finding accurate kinetic energies. Since we know that c0 Z 7/3 becomes exact in
a relative sense as N = Z → ∞, we define ∆TS = TS − c0 Z 7/3 and investigate ∆TS as a
function of Z. How accurate is the asymptotic expansion for ∆TS ? In Fig. 4.1, we evaluate
TS for atoms (see section 4.3 for details) and plot the percentage error in ∆TS , for all atoms
and the asymptotic series with just two terms. The series is incredibly accurate, with only
a 13% error for N=2 (He), and 14% for N=1. Thus, any approximation that reproduces
the correct asymptotic series (up to and including the c2 term) is likely to produce a highly
% errors in asymptotic series
accurate TS .
20
∆Ts = Ts - c0 Z
7/3
10
0
-10
-20
0
10
20
Z
30
40
Figure 4.1: Percentage error between c1 Z 2 + c2 Z 5/3 and ∆TS = TS − c0 Z 7/3 .
To demonstrate the power and the significance of this approach, we apply it directly to the
29
first term (where the answer is already known, but perhaps not yet fully appreciated in the
DFT community). Using any (all-electron) electronic structure code, one calculates the total
energies of atoms for a sequence running down a column in the periodic table. By sticking
with a specific column, one reduces the oscillatory contributions across rows, and the alkaliearth column yields the most accurate results. By then fitting the resulting curve of TS /Z 7/3
as a function of Z −1/3 to a parabola, one finds c0 = 0.7705. Now assume one wishes to make
a local density approximation (LDA) to TS , but knows nothing about the uniform electron
gas. Dimensional analysis (coordinate scaling) yields (51)
T
loc
[n] = AS I,
I=
Z
d3 r n5/3 (r) ,
(4.5)
but does not determine the constant, AS . A similar fitting of I, based on the corresponding
self-consistent densities, gives a leading term of 0.2677 Z 7/3 , yielding AS = 2.868. Thus we
have deduced the local approximation to the non-interacting kinetic energy.
A careful inspection of the above argument reveals that the uniform electron gas is never
mentioned. As N grows, the wavelength of the majority of the particles becomes short
relative to the scale on which the potential is changing, loosely speaking, and semiclassical
behavior dominates. The local approximation is a universal semiclassical result, which is
exact for a uniform gas simply because that system has a constant potential. On the basis
of that argument, we know AS =(3/10)(3π 2)2/3 = 2.871, demonstrating that (for this case)
our result is accurate to about 0.1%. This argument tells us that the reliability of the local
approximation is no indicator of how rapidly the density varies. That this argument is correct
for neutral atoms was carefully proven by Lieb and Simon in 1973 (52) and later generalized
by Lieb to all matter (46).
The focus of the first part of this paper is on the remaining two known coefficients (c1
and c2 ) and how well the GEA performs for them. We evaluate those gradient terms by
30
fitting asymptotic series and find that the traditional gradient expansion does well, but is
not exact. From this information, we develop a modified gradient expansion approximation
that reproduces the correct asymptotic coefficients c1 and c2 , merely as an illustration of
the power of asymptotic exactness. We test it on a variety of systems, finding the expected
behavior.
In Section 4.5, we present a parametrization of the TF density which is more accurate
than previous parametrizations. The TF density has a simple scaling with Z and becomes
relatively exact and slowly-varying for a neutral atom as Z → ∞, breaking down only near
the nucleus and in the tail. We compare various quantities of our parametrization with exact
values and earlier parametrizations, and analyze the properties of the TF density.
4.3
Large Z Methodology
We begin with a careful methodology for extracting the asymptotic behavior from highly
accurate numerical calculations. Fully numerical DFT calculations were performed using the
OPMKS code (38) to calculate the total energies of neutral atoms using ‘exact exchange’.
This is simply minimizing the Hartree-Fock energy, subject to the constraint of a multiplicative potential (18). The spin-density functional version of TS has been used for all systems
(53). We refer throughout to these as the KS results, and none of our analysis depends on
which approximation we use. The coefficients c0 , c1 and c2 are the same over a wide range
of approximations from exact exchange-correlation to local-density exchange.
To attain maximum accuracy for c1 and c2 , we need to suppress the oscillations which come
at the same order as the next term, c3 Z 4/3 . Consider first the KS results (TS ). We investigate
the differences between TS /Z 7/3 and c0 + c1 Z −1/3 + c2 Z −2/3 in Fig. 4.2. We extract 6 data
points (Z=24 (Cr), 25 (Mn), 30 (Zn), 31 (Ga), 61 (Pm), and 74 (W)) which have the
31
0.04
0.03
0.02
∆Ts
0.01
0
-0.01
-0.02
-0.03
-0.04
0
0.2
0.4
0.6
0.8
1
-1/3
Z
Figure 4.2: Difference between TS /Z 7/3 and c0 + c1 Z −1/3 + c2 Z −2/3 as a function of Z −1/3 .
smallest differences, i.e., nearest to where the curve crosses the horizontal axis. We then
make a least-squares fit with a parabolic form in Z −1/3 , ignoring the oscillation term,
TS
= 0.768745 + c1 Z −1/3 + c2 Z −2/3 .
Z 7/3
(4.6)
Effectively, we solve two linear equations for c1 and c2 . We explicitly include c0 = 0.768745,
since we don’t have enough data points to extract c0 accurately, especially in the region
Z −1/3 < 0.2. It is important to control the behavior of the fitting line at Z → ∞. This
fitting yields an accurate estimate of c1 = −0.5000 and c2 = 0.2702, with error less than 1%,
demonstrating the accuracy of our method for c1 and c2 .
We repeat the same procedure to extract c1 and c2 coefficients of TF and second- and fourthorder GEA’s which are given by
T GEA2 = T TF + T (2) ,
(4.7)
and (54; 41; 55):
T GEA4 = T TF + T (2) + T (4) .
(4.8)
32
These gradient corrections to the local approximation are given by
T
(2)
T
(4)
5
=
27
Z
8
=
81
Z
d3 r τ TF (r)s2 (r) ,
(4.9)
and
3
d rτ
TF
9
s4 (r)
2
2
(r) q (r) − q(r)s (r) +
,
8
3
(4.10)
where τ TF (r), s(r), and q(r) are defined as
τ TF (r) =
3 2
k (r)n(r) ,
10 F
(4.11)
s(r) =
|∇n(r)|
,
2kF(r)n(r)
(4.12)
q(r) =
∇2 n(r)
,
4kF2 (r)n(r)
(4.13)
and kF (r) = (3π 2 n(r))1/3 .
We have also applied this procedure to both T (2) and T (4) . Since the asymptotic expansions
of these energies begin at Z 2 , we extract only a c1 and a c2 for each using the following
equations:
T GEA2 − T TF
= ∆c1 Z −1/3 + ∆c2 Z −2/3 ,
7/3
Z
T GEA4 − T GEA2
= ∆c1 Z −1/3 + ∆c2 Z −2/3 .
Z 7/3
33
(4.14)
These results are also included in Table 4.1, and are of course consistent with our results
from Eq. (4.6).
Table 4.1: The coefficients in the asymptotic expansion of the KS kinetic energy and various
local and semilocal functionals. The fit was made to Z=24 (Cr), 25 (Mn), 30 (Zn), 31 (Ga),
61 (Pm), and 74 (W). The functionals of the last two rows are defined in section 4.4.
Exact
TS
T TF
T (2)
T (4)
T GEA2
T GEA4
T GGA a
T LmGGA a
a
4.4
c1
-0.5000
-0.5000
-0.6608
0.1246
0.0162
-0.5362
-0.5200
-0.5080
-0.5089
c2
0.2699
0.2702
0.3854
-0.0494
0.0071
0.3360
0.3431
0.2918
0.3174
See section 4.4
Results and Interpretation
To understand the meaning of the above results, begin with the values of c1 . We have combined the results of the T (2) and T (4) fits with that of the T TF fit to produce the asymptotic
coefficients of T GEA2 and T GEA4 . We check that these combinations produce the same coefficients in Table 4.1 which are found from the direct fitting of T GEA2 and T GEA4 using Eq.
(4.6). The exact value of c1 is −1/2. We see that the local approximation (TF) gives a
good estimate, −0.66. Then the second-order gradient expansion yields −0.54, reducing the
error by a factor of 5. Finally, the fourth-order gradient expansion yields −0.52, a further
improvement, yielding only a 4% error in its approximation to the Scott correction (56).
For c2 , the gradient expansion is less useful. The exact result is 0.27, while the TF approximation overestimates this as 0.39. The GEA2 result is only slightly reduced (0.34), and the
34
fourth-order correction has the wrong sign.
To understand how important these results can be, we consider how exchange and correlation
functionals are constructed. Often, such constructions begin from the GEA, which is then
generalized to include (in an approximate way) all powers of a given gradient. For slowly
varying densities, it is considered desirable to recover the GEA result. But we have seen here
how this conflicts with the asymptotic expansion, and in Ref (45), it was shown how the
asymptotic expansion is more significant to energies of real materials, and how successful
GGA’s for atoms and molecules well-approximate the large-Z asymptotic result, not the
slowly-varying gas.
Atoms: To illustrate this point, we construct here a trivial modified gradient expansion,
MGEA2, designed to have the correct asymptotic coefficients, in so far as is possible. Thus
T MGEA2 = T TF + 1.290 T (2)
(4.15)
The enhancement coefficient has been chosen to make cMGEA2
= −1/2. In Table 4.2, we list
1
the results of several different approximations for the alkali-earth atoms. Because the GEA2
error passes through 0 around Z=8, its errors are artificially low.
35
Table 4.2: KS kinetic energy (T ) in hartrees and various approximations for alkali-earth atoms.
36
Atom
Z
TS
T TF
%err
T GEA2
%err
T MGEA2
%err
T GEA4
%err
T MGEA4
%err
Be
4
14.5724
13.1290
-10
14.6471
0.5
15.0880
3.5
14.9854
2.8
14.5453
-0.2
Mg
12
199.612
184.002
-8
198.735
-0.4
203.014
1.7
201.452
0.9
199.924
0.2
Ca
20
676.752
630.064
-7
672.740
-0.6
685.136
1.2
680.286
0.5
677.433
0.1
Sr
38
3131.53
2951.89
-6
3110.44
-0.7
3156.50
0.8
3136.76
0.2
3134.48
0.09
Ba
56
7883.53
7478.27
-5
7829.36
-0.7
7931.34
0.6
7886.19
0.03
7888.14
0.06
Ra
88
23094.3
22065.8
-4
22945.9
-0.6
23201.5
0.5
23083.9 -0.05
23110.5
0.07
We can repeat this exercise for the fourth order, matching both c1 and c2 to exact values.
Now we find:
T MGEA4 [n] = T TF[n] + 1.789 T (2) [n] − 3.841T (4) [n]
(4.16)
i.e., strongly modified gradient coefficients. This is somewhat arbitrary, as there are several
terms in T (4) , and there’s no real reason to keep their ratios the same as in GEA (Eq. (4.10)).
However, the results of Table 4.2 and Fig. 4.3 speak for themselves. The resulting functional
is better than either GEA for all the alkali-earths. Of course, TS is positive for any density,
as are the terms T TF , T (2) and T (4) of the GEA. Eq. (4.16) however can be improperly
× 100
6
OEP
4
OEP
2
app
negative for rapidly-varying densities, and so is not suitable for general use.
0
0.8
GEA, 2nd
0.6
GEA, 4th
MGEA, 2nd 0.4
MGEA, 4th 0.2
40
60
80
100
(T
-T
)/T
0
-0.2
20
-2
0
20
40
Z
60
80
Figure 4.3: Percentage errors for atoms (from Z = 1 to Z = 92) using various approximations.
Molecules: The improvement in total kinetic energies is not just confined to atoms. Also,
for non-interacting kinetic energies of molecules, using the data in Ref. (57), Eq. (4.16)
gives a better average of the absolute errors in hartree (0.6) than T TF (9.4), T GEA2 (0.9),
and T GEA4 (0.8), shown in Table 4.3. Of greater importance are energy differences. For
atomization kinetic energies, also using the data in Ref. (57), T TF gives the best averaged
absolute error (0.25), which is worsened by gradient corrections. Since the GEA does not have
37
Table 4.3: Non-interacting kinetic energy (in hartrees) for molecules, and errors in approximations. All values are evaluated on the converged KS orbitals and densities obtained with
B88-PW91 functionals, and the MGEA4 kinetic energies are evaluated using the TF and the
GEA data from Ref. (57).
Atom
H
B
C
N
O
F
H2
HF
H2 O
CH4
NH3
BF3
CN
CO
F2
HCN
N2
NO
O2
O3
MAE2
1
2
TS 1
0.500
24.548
37.714
54.428
74.867
99.485
1.151
100.169
76.171
40.317
56.326
323.678
92.573
112.877
199.023
92.982
109.013
129.563
149.834
224.697
T TF 1
-0.044
-2.506
-3.731
-4.993
-6.990
-9.093
-0.142
-9.016
-7.074
-3.773
-5.292
-29.052
-8.940
-10.694
-18.367
-8.925
-10.487
-12.342
-14.186
-21.636
9.364
T GEA2 1 T GEA4
0.011
0.032
-0.058
0.476
-0.154
0.600
-0.097
0.904
-0.546
0.765
-0.933
0.659
-0.014
0.033
-0.920
0.639
-0.692
0.565
-0.140
0.619
-0.400
0.587
-2.641
2.454
-0.687
0.978
-0.911
1.036
-2.201
0.925
-0.658
1.008
-0.916
0.999
-1.240
0.962
-1.527
0.965
-2.699
1.028
0.872
0.812
1
T M GEA4
-0.026
-0.177
-0.228
-0.078
-0.497
-0.609
-0.094
-0.520
-0.484
-0.189
-0.331
-1.370
-0.570
-0.670
-1.451
-0.534
-0.719
0.279
-1.110
-2.071
0.600
Ref. (57)
Mean absolute error
the right quantum corrections from the edges, turning points and Coulomb cores (10), GEA
does not improve on the atomization process. However, the TF kinetic energy functional
is always the dominant term. So, TF gives very good results on the atomization kinetic
energies. But the error (0.29) of Eq. (4.16) is smaller than that of T GEA2 (0.36) and T GEA4
(0.44). In either case, Eq. (4.16) works better for atoms and molecules than the fourth-order
gradient expansion. Thus, requiring asymptotic exactness is a useful and powerful constraint
in functional design.
38
Jellium surfaces: We test this MGEA4 functional for jellium surface kinetic energies.
As shown in Table 4.4, the T (4) term in T GEA4 improves the jellium surface kinetic energy
in comparison to the results of T GEA2 , but Eq. (4.16) worsens the jellium surface kinetic
energies due to the strongly modified coefficient of T (4) . This is a confirmation of our general
approach. By building in the correct asymptotic behavior for atoms, including the Scott
correction coming from the 1s region, we worsen energetics for systems without this feature.
Table 4.4: Jellium surface kinetic energies (erg/cm2 ) and % error, which is (σSapp − σSex )/σSex ,
of each approximation.
rs
2
4
6
a
b
Exact
-5492.7
-139.9
-3.4
T TF
11
54
660
T GEA2
2.5
22
330
T GEA4
1.1
11
180
T MGEA2
-0.9
12
238
a
T MGEA4
0.73
36
675
b
T LmGGA
1.3
15
280
See Eq. (4.15)
See Eq. (4.16)
Jellium spheres: We also investigate the kinetic energies of neutral jellium spheres (with
KS densities using LDA exchange-correlation and with rS = 3.9) from Ref. (58). The analysis
of the results is based upon the liquid drop model of Refs. (59; 60). We write
4
TS (rS , N) = πR3 τ unif (rS ) + 4πR2 σS + 2πRγSeff (rS , N),
3
(4.17)
where R is the radius of the sphere of uniform positive background. Since we know the
bulk (uniform) kinetic energy density, τ unif , and the surface kinetic energy σS for a given
functional, we can extract γSeff (rS , N) from this equation, and
lim γSeff (rS , N) = γS (rS )
(4.18)
N →∞
is the curvature energy of jellium. We calculate γSeff (rS , N) using the TF, GEA, MGEA,
39
and a Laplacian-level meta-GGA (LmGGA) of Ref. (58), which is explained further in the
following subsection. From Table 4.5, we observe that: (i) Gradient corrections in GEA
Table 4.5: 104 × (γSeff (rS , N) − γSTF (rS , N)) in atomic units vs. N = Z for neutral jellium
spheres with rS = 3.93 with various functionals. As N = Z → ∞, γSeff tends to the curvature
kinetic energy of jellium, γS .
N
2
8
18
58
92
254
a
b
Exact
-1.8
-1.9
-0.5
-0.8
-1.7
-0.5
T GEA2
1.1
1.0
1.2
1.3
1.2
1.4
T GEA4
2.4
2.1
2.0
2.2
2.0
2.3
T MGEA2
1.5
1.3
1.6
1.7
1.5
1.8
a
T MGEA4
-2.8
-2.3
-0.7
-1.1
-1.0
-0.9
b
T LmGGA
1.9
-5.1
-6.4
-3.2
-1.9
-
See Eq. (4.15)
See Eq. (4.16)
worsen γSeff . (ii) The LmGGA of Ref. (58) is even worse than T GEA4 . (iii) Eq. (4.15) (which
has the right c0 and c1 ) is not so good, but better than T GEA4 . (iv) Eq. (4.16) (which has
the right c0 , c1 , and c2 ) gives good results.
Existing approximations: We suggest that the large-Z asymptotic expansion is a necessary condition that an accurate kinetic energy functional should satisfy, but is not sufficient.
We show this by testing two kinds of semilocal approximations (GGA and meta-GGA) to
the kinetic energy functionals.
Recently, Tran and Wesolowski (61) constructed a GGA-type kinetic energy functional using the conjointness conjecture. They found the enhancement factor by minimizing mean
absolute errors of kinetic energies for closed-shell atoms. We evaluate the kinetic energies of
atoms using this functional (T GGA ) and extract the asymptotic coefficients shown in Table
4.1. This gives a good c1 coefficient (-0.51), with c2 (0.29) close to the correct value (0.27),
and so is much more accurate than the GEA’s.
Perdew and Constantin (58) constructed a LmGGA for the positive kinetic energy density τ
40
that satisfies the local bound τ ≥ τW , where τW is the von Weizsäcker kinetic energy density,
and tends to τW as r → 0 in an atom. It recovers the fourth-order gradient expansion in
the slowly-varying limit. We calculate the asymptotic coefficients shown in Table 4.1 for this
functional. These values are better than those of T GEA4 . The good c1 from T GEA4 appears
somewhat fortuitous, since there is nothing about a slowly-varying density that is relevant
to a cusp in the density. The good Scott correction c1 from the LmGGA comes from correct
physics: LmGGA recovers the von Weizsäcker kinetic energy density in the 1s cusp, without
the spurious but integrable divergences of the integrand of T GEA4 .
We finish by discussing other columns of the periodic table. We have also performed all
these calculations on the noble gases. In fact, from studies of the asymptotic series (62), it
is known that the shell-structure occurs in the next order, Z 4/3 , and that the noble gases
are furthest from the asymptotic curves. But Table 4.6 shows our functionals work almost
as well for the noble gas series.
41
Table 4.6: KS kinetic energy (T ) in hartrees and various approximations for noble atoms.
42
Atom
Z
TS
T TF
%err
T GEA2
%err
T MGEA2
%err
T GEA4
%err
T MGEA4
%err
He
2
2.86168
2.56051
-11
2.87847
0.6
2.97083
3.8
2.96236
3.5
2.80717
-1.9
Ne
10
128.545
117.761
-8
127.829
-0.6
130.753
1.7
129.737
0.9
128.447
-0.08
Ar
18
526.812
489.955
-7
524.224
-0.5
534.178
1.4
530.341
0.7
527.772
0.2
Kr
36
2752.04
2591.20
-6
2733.07
-0.7
2774.27
0.8
2756.72
0.2
2754.17
0.08
Xe
54
7232.12
6857.94
-5
7183.78
-0.7
7278.42
0.6
7236.65
0.06
7237.85
0.08
Rn
86
21866.7
20885.7
-4
21725.4
-0.6
21969.3
0.5
21857.2 -0.04
21881.7
0.07
4.5
Modern Parametrization of Thomas-Fermi Density
Our asymptotic expansion study gives new reasons for studying large Z atoms. Our approximate functionals were tested on highly accurate densities, but ultimately, self-consistency is
an important and more-demanding test. Any approximate functional yields an approximate
density via the Euler equation. In this section, we present a new, modern parametrization
of the neutral atom TF density, which is more accurate than earlier versions (63; 64).
The TF density of a neutral atom can be written as
Z2
n(r) =
4πa3
3/2
Φ
,
x
(4.19)
where a = (1/2)(3π/4)2/3 and x = Z 1/3 r/a, and the dimensionless TF differential equation
is
d2 Φ(x)
=
dx2
r
Φ3 (x)
,
x
Φ(x) > 0,
(4.20)
which satisfies the following initial conditions:
Φ(0) = 1 ,
Φ′ (0) = −B ,
B = 1.5880710226 .
(4.21)
We construct a model for Φ which recovers the first eight terms of the small-x expansion
and the leading term of the asymptotic expansion at large-x (Φ(x) → 144/x3 , as x → ∞).
√
Following Tal and Levy (65), we use y = x as the variable, because of the singularity of
the TF equation. Our parametrization is
Φmod (y) =
1+
9
X
p=2
αp y p
!
5
X
α9 y 15
/ 1 + y9
βp y p +
144
p=1
43
!
(4.22)
1
Φ (y)
0.8
0.6
0.4
0.2
0
0
1
2
3
y=x
4
5
1/2
Figure 4.4: Accurate numerical Φ(y) and parametrized Φ(y) can not be distinguished.
8e-06
ex
(y)-Φ (y)
4e-06
Φ
mod
0
-4e-06
-8e-06
0
2
4
6
8
10
12
14
1/2
y=x
Figure 4.5: Errors in the model, relative to numerical integrations.
44
where αi and βi are coefficients given in the Table 4.7. The values of αi are fixed by the small
y-expansion, while those of βi are found by minimization of the weighted sum of squared
residuals, χ2 , for 0 < y < 10. The χ2 was minimized using the Levenberg-Marquardt method
(66). This method is for fitting when the model depends nonlinearly on the set of unknown
parameters. 1000 points were used, equally spaced between y = 0 and y = 10. We plot
Table 4.7: The values of βi are found by fitting Eq.(4.22) to the accurate numerical solution,
and those of αi are the parameters of small-y expansion (65). B is given by 1.5880710226.
α2
α3
α5
α6
α7
α8
α9
−B
4/3
−2B/5
1/3
3B 2 /70
−2B/15
2/27 + B 3 /252
β1
β2
β3
β4
β5
−0.0144050081
0.0231427314
−0.00617782965
0.0103191718
−0.000154797772
the accurate Φ(y) and our model in Fig. 4.4, and the differences between them in Fig. 4.5.
These graphs illustrate the accuracy of our parametrization.
In Table 4.5 we calculate several moments using our model and existing models that were
proposed by Gross and Dreizler (63) and Latter (64). The Latter parametrization is
ΦL (x) = 1/(1 + 0.02747x1/2 + 1.243x − 0.1486x3/2 + 0.2303x2 + 0.007298x5/2
+0.006944x3) ,
(4.23)
and the Gross-Dreizler model (which correctly removes the
√
x term) is:
ΦGD (x) = 1/(1 + 1.4712x − 0.4973x3/2 + 0.3875x2 + 0.002102x3) .
(4.24)
Lastly, we introduce an extremely simple model that we have found useful for pedagogical
45
purposes (even when N differs from Z). We write
nped (r) =
N
αN 2/3
1 −r/R
e
,
R
=
,
2π 3/2 R3/2 r 3/2
Z − βN
(4.25)
√ √
where α = (9/5 5)( 3π/4)1/3 and β = 1/2 − 1/π have been found from integration of the
TF kinetic and Hartree energies, respectively, and R minimizes the TF total energy. For
N = Z, this yields:
ped
Φ
−2a(1−β)x/3α
(x) = γ e
√ 5 5 1 1
, γ= √
.
+
6 3 2 π
(4.26)
This crude approximation does not satisfy the correct initial conditions of Eq. (4.21):
Φped (0) = γ = 0.880361 (6= 1) ,
Φped′ (0) = −
125(2 + π)2
= −0.48 (6= −1.59) .
648(4π 5 )1/3
(4.27)
To compare the quality of the various parametrizations, we calculate the p-th moment of the
j-th power of Φ(x)/x:
(p)
Mj
=
Z
p
dx x
Φ(x)
x
j
.
(4.28)
Many quantities of interest can be expressed in terms of these moments:
1) Particle number: To ensure
R
d3 r n(r) = N, we require
(2)
M3/2 = 1
(4.29)
46
2) TF kinetic energy: The TF kinetic energy is c0 Z 7/3 , which implies
5
(2)
M5/2 = B .
7
3) The Hartree energy is U =
(4.30)
1
2
RR
′
)
d3 r d3 r ′ n(r)n(r
=
|r−r′ |
(1)
1
M3/2 Z 7/3 ,
7a
which implies
(1)
M3/2 = B .
(4.31)
R
(1)
4) The external energy is defined as Vext = − d3 r Z n(r)/r = − a1 M3/2 Z 7/3 for the exact TF
density, which also implies Eq. (4.31).
5) The local density approximation exchange energy is defined as EXLDA = AX
1
(2)
R
dr n4/3 (r),
where AX = −(3/4)(3/π)1/3 , so for TF, EXLDA = AX (4πa3 )− 3 M2 Z 5/3 , which implies
(2)
M2 = 0.615434679 ,
(4.32)
extracted from our accurate numerical solution. LDA exchange suffices (43; 45) for asymptotic exactness to the order displayed in Eqs. (4.3) and (4.4); for a numerical study, see Ref.
(67; 68).
47
(p)
Table 4.8: Various moments calculated with our model and with the models of Ref. (63; 64). Here Mj
j
R
dx xp Φ(x)
x
our model
% err
Gross and Dreizler (63)
% err
Latter(64)
% err
Φped (x)
% err
exact
(2)
0.999857885
-0.01
1.008
0.8
0.999
-0.04
1
0
1
(2)
1.13426462
-0.006
1.1299
-0.4
1.137
0.2
1.11
-2
5B/7
(2)
0.615438208
0.001
0.6129
-0.4
0.616
0.02
0.72
16
0.6154346791
(1)
1.58799857
-0.005
1.5844
-0.2
1.589
0.07
1.62
2
B
moment
48
M3/2
M5/2
M2
M3/2
1
is given by
Numerical result from the TF differential equation.
Table 4.5 shows that our modern parametrization is far more accurate than existing models
by all measures, and that our simple pedagogical model is roughly correct for many features.
Finally, we make some comparisons with densities of real atoms to illustrate those features
of real atoms that are captured by TF. The radial density, s(r) (Eq. (4.12)), and q(r) (Eq.
(4.13)) are given by
4πr 2 n(r) = Z 4/3 f (x)/a ,
where f (x) =
√
(4.33)
x Φ3/2 (x),
s(r) =
a1 |g(x)|
, a1 = (9/2π)1/3 /2 ,
Z 1/3 f (x)
(4.34)
q(r) =
a21 {g 2 (x) + 2x2 Φ(x)Φ′′ (x)}
,
3Z 2/3
f 2 (x)
(4.35)
and
where g(x) is defined as Φ(x) − xΦ′ (x). The gradient relative to the screening length is
t(r) =
p
|∇n(r)|
, where kS (r) = 4kF (r)/π ,
2kS (r)n(r)
(4.36)
a2 |g(x)|
35/6 π 1/3
√ = 0.6124 .
,
a
=
2
(x3 Φ5 (x))1/4
28/3 a
(4.37)
and here
t(r) =
We also show large- and small-x limit behaviors of various quantities using Φ(x) → 144/x3
49
as x → ∞ and Φ(x) → 1 − Bx + · · · as x → 0.
Z2 1
4πa3 x3/2
Z 4/3 √
x
a
a1 1
√
Z 1/3 x
a21 1
3Z 2/3 x
a2
x3/4
x→0
←−
x→∞
n(r)
−→
x→0
x→∞
←− 4πr 2 n(r) −→
←−
x→0
s(r)
x→∞
←−
x→0
q(r)
x→∞
x→0
t(r)
x→∞
←−
−→
−→
−→
432Z 2
,
a3 πx6
144Z 4/3
,
ax5/2
a1 x
,
3Z 1/3
5a21 x2
,
54Z 2/3
2a2
√ .
3
(4.38)
(4.39)
(4.40)
(4.41)
(4.42)
0.5
Ba
Ra
TF
4πr n(r) / Z
4/3
0.4
2
0.3
0.2
0.1
0
0
1
2
3
1/3
Z r = ax
Figure 4.6: Plot of the scaled radial densities of Ba and Ra using Eq.(4.33) and SCF densities.
TF scaled densities of Ba and Ra are on top of each other.
We plot the Z-scaled accurate self-consistent densities and TF radial densities of Ba (Z = 56)
and Ra (Z = 88) in Fig. 4.6. Although the shell structure is missing, and the decay at a
large distance is wrong, the overall shape of the TF density is relatively correct.
In Figs. 4.7, 4.8, and 4.9, we plot the scaled s(r), q(r), and t(r) using the self-consistent
and TF densities of Ba and Ra. In particular, t(r) measures how fast the density changes on
50
3.5
Ba
Ra
TF
3
1/3
Z s(r)
2.5
2
1.5
1
0.5
0
1
0.5
1.5
1/3
Z r = ax
Figure 4.7: Plot of the scaled reduced density gradient s(r) (relative to the local Fermi
wavelength) vs. Z 1/3 r.
the scale of the TF screening length, and its magnitude does not vary with Z in TF theory.
From these figures, we see that s(r), q(r) and t(r) of the TF density diverge near the nucleus,
since the TF density does not satisfy Kato’s cusp condition.
8
Ba
Ra
TF
4
2/3
Z q(r)
6
2
0
-2
0
1
0.5
1.5
2
1/3
Z r = ax
Figure 4.8: Plot of the scaled reduced Laplacian q(r) (relative to the local Fermi wavelength)
vs. Z 1/3 r.
When N = Z → ∞ for a realistic density, s(r) is small except in the density tail (s ∼ Z −1/3
over most of the density), and q(r) is small except in the tail and 1s core regions (q ∼ Z −2/3
over most of the density). This is why gradient expansions for the kinetic and exchange
energies, applied to realistic densities, work as well as they do in this limit. The kinetic and
51
exchange energies have only one characteristic length scale, the local Fermi wavelength, but
the correlation energy also has a different one, the local screening length. Since t(r) is not
and does not become small in this limit, gradient expansions do not work well at all for the
correlation energies of atoms (45). The standard of “smallness” for s and q, and the more
severe standard of smallness for t, are explained in Refs. (45) and (69; 70).
Finally we evaluate T (0) + T (2) on the TF density. We find the correct c0 in the Z → ∞
expansion from T (0) , but c1 vanishes, due to the absence of a proper nuclear cusp, and c2
diverges because T (2) diverges at its lower limit of integration.
4.6
Summary
We have shown the importance of the large-N limit for density functional construction of the
kinetic energy (with the functional evaluated on a Kohn-Sham density), and also provided a
modern, highly accurate parametrization of the neutral-atom TF density. Our results should
prove useful in the never-ending search for improved density functionals.
4
Ba
TF
Ra
t(r)
3
2
1
0
0
1
2
3
1/3
Z r = ax
Figure 4.9: Plot of the reduced density gradient t(r) (relative to the local screening length)
vs. Z 1/3 r. As r → ∞, the TF t → 0.7071.
52
For atoms and molecules, the large-N limit seems more important than the slowly-varying
limit. On the ladder (71) of density-functional approximations, there are three rungs of
semilocal approximations (followed by higher rungs of fully nonlocal ones). The LDA uses
only the local density, the GGA uses also the density gradient, and the meta-GGA uses in
addition the orbital kinetic energy density or the Laplacian of the density. For the exchangecorrelation energy, the GGA rung cannot (45; 69) simultaneously describe the slowly-varying
limit and the N = Z → ∞ limit for an atom, and we have found here that the same is true
(but less severely by percent error of a given energy component) for the kinetic energy. This
follows because, as N = Z → ∞, the reduced gradient s(r) of Eq. (4.12) becomes small over
the energetically important regions of the atom, as can be inferred from Fig. 4.7, so that
a GGA reduces to its own second-order gradient expansion even in regions where a metaGGA does not (45) (e.g., near a nucleus, where q(r) diverges but s(r) does not, as shown in
Figs. 4.7 and 4.8). For the kinetic as for the exchange-correlation energy, meta-GGA’s (58)
can recover both the slowly-varying and large-Z limits; it remains to be seen how well fully
nonlocal approximations (6; 72) can do this.
53
Chapter 5
Semiclassical Orbital-Free
Potential-Density Functional Theory
In this chapter, we discuss how to derive a uniform semiclassical approximation to the density
and kinetic energy (KE) density as a functional of the potential from the semiclassical Green’s
function. we apply this semiclassical scheme to 1D systems with two turning points and
compare the semiclassical result to the exact answer. Also, we find that this scheme can be
applied to 3D spherical systems with two turning points.
5.1
Uniform semiclassical density
Recently, we derived a semiclassical approximation to the density and KE density for systems
with hard wall boundaries(10). This has been further discussed in Ref. (73). The original
idea of a semiclassical Green’s function approach comes from Ref. (74).
Imagine a potential v(x) in 1D with hard walls at x = 0 and x = L. If the orbital energy
is greater than the maximum of the bottom potential, the following WKB wavefunction
54
satisfies the boundary conditions:
ψ(x) ≃ p
1
k(x)
sin θ(x) ,
(5.1)
p
where the semiclassical momentum k(x) = 2(E − v(x)) and the semiclassical phase θ(x) =
Rx
dx k(x). Hence, we can construct the semiclassical Green’s function from WKB wave0
functions using Eq. (2.18):
g(x; E) =
cos θ(L) − cos (2θ(x) − θ(L))
.
k(x) sin θ(L)
(5.2)
The Fermi energy (or chemical potential) EF is determined by the quantization condition
Z
0
L
dx
p
1
π,
2(EF − v(x)) = N +
2
(5.3)
where N is the number of particles, and EF lies between the highest occupied and the lowest
unoccupied levels.
Figure 5.1: Contour of integration in the complex E-plane.
Accounting for the requirement that EN ≤ EF ≤ EN +1 we pick the contour C shown in
Fig. 5.1: a vertical line along E = EF + iη connected to a semicircle, which encloses the
55
N lower-lying energy-eigenvalues E1 , . . . , EN . By choosing this particular shape for C we
ensure that the lowest |E| used is EF .
Via Eq. (2.19) where the contour C is chosen to pass through the real axis at EF in Fig.
5.1, the semiclassical density for N particles is obtained
nsemi (x) =
sin θF (x)
kF (x)
−
,
π
2TF kF (x) sin α(x)
(5.4)
where α(x) = πτF (x)/TF , the semiclassical traveling time from 0 to x τF (x) =
Rx
0
dx 1/kF (x),
and τF (L) = TF . Note that subscript F denotes evaluation of the given quantity at EF . This
semiclassical approximation generally gives very accurate densities (10; 73) for arbitrary
bottom potentials.
However, the WKB approximation is not a good choice for construction of a uniform semiclassical Green’s function for 1D systems with turning points, since the WKB wavefunction
diverges at turning points due to the 1/k(x) factor. we use a uniform semiclassical approximation instead of WKB to avoid this problem.
5.1.1
The leading term
We construct the Green’s function for a single turning point by approximating ψl,r (x) by
means of the uniform semiclassical solutions (12; 75)
θ1/6 (x)
ψl (x) = p
Ai(−z(x)) ,
k(x)
θ1/6 (x)
unif
Ai(−e2πi/3 z(x)) ,
ψr (x) = p
k(x)
unif
(5.5)
(5.6)
56
p
where the semiclassical momentum k(x) = 2(E − v(x)), the semiclassical phase θ(x) =
Rx ′
dx k(x′ ) , z(x) = (3θ(x)/2)2/3 , and a is the turning point on the left side. In this
a
approximation, the Wronskian becomes
1/3
1 5πi/6
2
e
.
W (E) =
3
2π
(5.7)
In atomic units, p = 1/2 in Eq. (2.18), our semiclassical approximation to the diagonal
Green’s function for a single turning-point problem becomes
semi
gST
p
2π z(x)
(x, E) =
Ai(−z(x))[Ai(−z(x)) − i Bi(−z(x))] .
i k(x)
(5.8)
As the number of particles N → ∞, EF ≫ η for the dominant contributions to the integral,
allowing us to expand all quantities in powers of the imaginary part of the energy η
1
1
=
k(x; EF + iη)
kF (x)
iη
1 − 2 + ...
kF
,
θ(x; EF + iη) = θF (x) + iητF (x) + . . . ,
Θ(EF + iη) = ΘF + iηTF + . . . ,
(5.10)
(5.11)
τF (x)
+ ... .
z(x; EF + iη) = zF (x) + iη p
zF (x)
Here, we introduce the quantities τF (x) =
(5.9)
Rx
aF
(5.12)
dx′ 1/kF(x′ ), the classical time for a particle at
EF to travel from aF to x, and TF = τF (bF ). aF and bF are the left and right turning points
determined by EF . Keeping terms up to first order in η and using the integral representation
57
of Airy functions (76), Eq. (2.19) yields
Z
∞
"√
zF
+ iη
kF
τF (x)
zF (x)
p
idη
− 3
kF (x) zF (x) kF (x)
0
Z ∞
√
1
× 3
dt e−if (t)−tητF (x)/ zF (x) .
2π 0
1
nsemi
(x) = − ℜ
ST
π
!#
(5.13)
We make use of the semiclassical quantization condition for the two-turning-point problem,
which states that Θ/π is an integer +1/2, i.e., ΘF = Nπ. Direct evaluation of the integration
with respect to η gives
nsemi
(x) =
ST
2zF (x)
[zF (x)Ai2 (−zF (x)) + Ai′ (−zF (x))2 ] .
kF (x)τF (x)
(5.14)
However, this semiclassical density for a single turning point system cannot recover the
Thomas-Fermi (TF) density at the asymptotic limit, since the uniform semiclassical wavefunction does not know about the existence of another turning point. Hence, to recover the
TF density, we set the traveling time τF (x) as
τF (x) =
3θF (x)
.
kF2 (x)
(5.15)
This yields
kF (x)
[zF (x)Ai2 (−zF (x)) + Ai′ (−zF (x))2 ] .
nsemi
(x) = p
ST
zF (x)
(5.16)
Using the asymptotic form of Airy functions(76), we obtain from Eq. (5.16)
nsemi
ST (x) ∼
kF (x) kF (x) cos(2θF (x))
−
,
π
6πθF (x)
(5.17)
which recovers the TF density as the leading term.
58
5.1.2
Quantum corrections
The Green’s function(74) for a two-turning-point system is given as
g
semi
p
z(x)
[tan Θ Ai2 (−z(x)) + Ai(−z(x))Bi(−z(x))] .
(x; E) = −2π
k(x)
(5.18)
semi
It can be split into gST
(x; E) (Eq. (5.8)) and g QC (x; E), which arises due to the other turning
point. So,
semi
g semi (x; E) = gST
(x; E) + g QC (x; E) ,
(5.19)
where
p
z(x)
1
g QC (x; E) = 4πi
Ai2 (−z(x)) .
k(x) 1 + e−2iΘ
(5.20)
Hence, the density becomes
semi
n
I
1
(x) =
dE g semi (x; E)
2πi C(µ)
I
1
semi
dE [gST
(x; E) + g QC (x; E)]
=
2πi C(µ)
QC
= nsemi
(x) ,
ST (x) + n
(5.21)
where nsemi
(x) is given in Eq. (5.16). The density nQC (x) is
ST
QC
n
I
p
z(x)
(x) =
dE
k(x)
C
2
1 + e−2iΘ
Ai2 (−z(x)) .
59
(5.22)
By taking E → EF + i η, expanding terms to the first order in η, and using the contour in
Fig. 5.1, nQC (x) becomes
QC
n
" p
zF (x)
(x) = 4ℜ i
kF (x)
Z
∞
0
Ai2 (−zF (x) − iητF (x)/
dη
1 + e2ηTF
p
zF (x))
#
.
(5.23)
Using the integral representation of the Airy function, nQC (x) can be expressed as an integral
representation
√
"Z
#
p
Z ∞ −ηtτF (x)/√zF (x)
∞
ηtτF (x)/ zF (x)
z
(x)
2
−
e
dt
e
F
√ sin f (t)
nQC (x) = 3/2
dη , (5.24)
2π
kF (x)
1 + e2ηTF
t
0
0
where f (t) = t3 /12 − zF (x)t + π/4. The integration with respect to η can be expressed as
Z
∞
0
√
√
p
zF (x)
π
e−ηtτF (x)/ zF (x) − eηtτF (x)/ zF (x)
dη =
−
csc
2ηT
F
1+e
tτF (x)
2TF
By setting t = 2
p
!
πτF (x)
p
t .
2TF zF (x)
(5.25)
zF (x) via the stationary phase approximation, the above integration is
simplified to yield
Z
0
∞
1
π
e−2ητF − e2ητF
dη =
−
csc
2ηT
F
1+e
2τF 2TF
πτF
TF
.
(5.26)
Hence, the density nQC (x) is
nQC (x) =
=
p
p
!
Z ∞
zF (x)
π zF (x)
dt
πτF (x)
1
−
csc
sin(f (t)) √
3/2
kF (x)τF (x)
kF (x)TF
TF
2π
t
0
!
p
p
π zF (x)
zF (x)
Ai(−zF (x))Bi(−zF (x)) .
−
kF (x)τF (x) kF (x)TF sin(α(x))
60
(5.27)
By using the asymptotic forms of Ai and Bi (76),
1
p
sin(2θF (x) + π/2)
2π zF (x)
1
p
=
cos(2θF (x)) ,
2π zF (x)
Ai(−zF (x))Bi(−zF (x)) ∼
(5.28)
Eq. (5.27) becomes
nQC (x) =
cos(2θF (x))
cos(2θF (x))
−
,
2πkF(x)τF (x) 2kF (x)TF sin(α(x))
(5.29)
and Eq. (5.15) gives the quantum correction term corresponding to Eq. (5.16):
nQC (x) =
cos(2θF (x))
kF (x) cos(2θF (x))
−
.
6πθF (x)
2kF(x)TF sin(α(x))
(5.30)
Combining Eq. (5.17) and Eq. (5.30) reproduces the asymptotic density formula in Ref.
(74):
nKS (x) =
5.1.3
kF (x)
cos(2θF (x))
−
.
π
2kF (x)TF sin(α(x))
(5.31)
The evanescent region
In the evanescent region (E − v(x) < 0), the uniform semiclassical wavefunctions are
|θ(x)|1/6
unif
ψl (x) = p
Ai(|z(x)|) ,
|k(x)|
|θ(x)|1/6
unif
Ai(e2πi/3 |z(x)|) .
ψr (x) = p
|k(x)|
(5.32)
(5.33)
61
Using the above wavefunctions, the Green’s function for a single turning-point system in this
region can be constructed:
semi
gST
p
2π |z(x)|
(x, E) =
Ai(|z(x)|)[Ai(|z(x)|) − i Bi(|z(x)|)] .
i |k(x)|
(5.34)
Using the same derivation, we find nsemi
(x) in the evanescent region to be
ST
|kF (x)|
nsemi
(x) = − p
[|zF (x)|Ai2 (|zF (x)|) − Ai′ (|zF (x)|)2 ] .
ST
|zF (x)|
(5.35)
The Green’s function in the evanescent region for two-turning-point systems is constructed:
g semi (x; E) = −2π tan Θ
p
|z(x)| 2
[Ai (|z(x)|) − iAi(|z(x)|)Bi(|z(x)|)] .
|k(x)|
(5.36)
Following the logic of Eq. (5.21), we can extract g QC (x; E) in this region as
p
|z(x)|
1
[Ai2 (|z(x)|) − iAi(|z(x)|)Bi(|z(x)|)] .
g QC (x; E) = 4πi
|k(x)| 1 + e−2iΘ
(5.37)
We then derive the quantum correction in the evanescent region, using the contour in Fig.
5.1 and find
nQC (x) =
5.2
e−2|θF (x)|
|kF (x)|e−2|θF (x)|
−
.
4|kF(x)|TF sinh(|α(x)|)
12π|θF (x)|
(5.38)
Uniform kinetic energy density
Using the definition of Eq. (2.20) and Green’s functions Eqs. (5.8) and (5.20), we derive a
semiclassical approximation to the KE density in the traveling and evanescent regions. We
62
apply the same method as in deriving the density with the only difference that we expand
the semiclassical quantities in the integrand up to the second order of η. Without going into
any details of the derivations, we show the final results.
In the traveling region, the leading KE density term is given as
tsemi
(x) =
ST
k 3 (x)
pF
[zF (x)Ai2 (−zF (x)) + Ai′ (−zF (x))2 ]
2 zF (x)
k 3 (x)
[2z 2 (x)Ai2 (−zF (x)) − Ai(−zF (x))Ai′ (−zF (x))
− F
9θF (x) F
+2zF (x)Ai′2 (−zF (x))] ,
(5.39)
which recovers the TF KE density (kF3 (x)/6π) at the asymptotic limit. The first order
quantum correction to the kinetic energy density is given by
tQC
(1) (x) =
kF3 (x) cos(2θF (x)) kF (x) cos(2θF (x))
−
.
12πθF (x)
4TF sin(α(x))
(5.40)
The second order KED quantum correction is given by
QC
t(2) (x) = −
p
zF (x)π 2 cos(α(x))
−
2
5/2
8zF (x) 2kF(x)TF2 sin (α(x))
kF3 (x)
!
Ai2 (−zF (x)) .
(5.41)
We now find analogous quantities in the evanescent region. The leading term of the KE
density is
tsemi
=
ST (x)
|kF (x)|3
p
[|zF (x)|Ai2 (|zF (x)|) − Ai′2 (|zF (x))|)]
2 |zF (x)|
|kF (x)|3
−
[2|zF (x)|2 Ai2 (|zF (x)|)
9|θF (x)|
−Ai(|zF (x)|)Ai′ (|zF (x)|) − 2|zF (x)|Ai′2 (|zF (x)|)] .
63
(5.42)
If we take the asymptotic forms of Ai and Ai′ , tsemi
ST (x) becomes zero at the asymptotic limit
in the evanescent region. The quantum corrections in this region are
tQC
(1) (x) =
|kF(x)|3 e−2|θF (x)|
|kF (x)|e−2|θF (x)|
−
,
24π|θF(x)|
8TF sinh(|α(x)|)
(5.43)
and
tQC
(2) (x) =
!
p
|zF (x)|π 2 cosh(|α(x)|)
|kF (x)|3
−
Ai2 (|zF (x)|) ,
8|zF (x)|5/2 2|kF(x)|TF2 sinh2 (|α(x)|)
(5.44)
giving a set of clearly analogous expressions for the KE density in both regions.
5.3
Different limit
However, the above quantum corrections result in a discontinuity at the turning point. At
the turning point, aF , the magnitude of Eq. (5.30) is twice that of Eq. (5.38). For a harmonic
oscillator (v(x) = x2 /2), Eq. (5.38) in the evanescent region is more accurate than Eq. (5.30)
in the traveling region at aF . (See Section 5.5.1 and Table 5.1 for the detailed discussion and
the data.)
To make these equations continuous, we define the following two points, xI and xII , in the
traveling region (see Fig. 5.2):
Z
Z
xI
aF
bF
xII
dx′ kF (x′ ) =
π
4
dx′ kF (x′ ) =
π
.
4
(5.45)
64
1.5
Potential
Density
EF
n(x)
1
xI
0.5
xII
aF
0
-3
-2
bF
-1
0
1
2
3
x
Figure 5.2: 1D systems.
At these xI and xII , nQC (x) (Eq. (5.30)) becomes zero since cos (2θF (xI/II )) = cos (π/2) = 0.
Only the leading term, nsemi
ST (x), remains. So we can divide the region to make the density
continuous.
With this separation of space, we obtain a continuous quantum corrections to the density:

cos(2θF (x))
kF (x) cos(2θF (x))


−
, for xI < x < xmid


6πθF (x)
2kF (x)TF sin(α(x))



 k (x) cos(2θ (x))
cos(2θF (x))
F
F
QC
−
, for aF < x < xI
n (x) =
12πθF (x)
4kF (x)TF sin(α(x))





e−2|θF (x)|
|kF(x)|e−2|θF (x)|


−
, for x < aF ,

4|kF (x)|TF sinh(|α(x)|)
12π|θF (x)|
(5.46)
where the mid-phase point, xmid , is determined by
Z
xmid
dx′ θF (x′ ) =
aF
Nπ
.
2
(5.47)
These formulas give a continuous density although discontinuities in the first derivative exist
at xI , xII and xmid .
65
Using the same argument, the tQC (x) can also be divided into three regions:

QC


tQC

(1),a + t(2),a , for xI < x < xmid



1 QC
QC
t (x) =
t(1),a + tQC
(2),a , for aF < x < xI

2




tQC + tQC , for x < aF .
(1),e
(2),e
5.4
(5.48)
Summary of equations
In this section, we summarize all previous results related to the semiclassical density and
KE density that will be used in this paper. We define the following semiclassical quantities
at EF .
The semiclassical momentum:
kF (x) =
p
2|EF − v(x)| .
(5.49)
The semiclassical phase:
θF (x) =
Z
x
kF (x′ )dx′ ,
(5.50)
aF
zF (x) = Sgn(θF (x))
3
|θF (x)|
2
2/3
.
(5.51)
The classical traveling time:
τF (x) =
Z
x
aF
1
dx′ .
′
kF (x )
(5.52)
66
For the semiclassical density (nsemi
and nQC (x)):
ST
kF (x)
p
nsemi
[zF (x)Ai2 (−zF (x)) + Ai′ (−zF (x))2 ] ,
ST (x) = ±
±zF (x)

kF (x) cos(2θF (x))
cos(2θF (x))


−
, for xI < x < xmid


6πθF (x)
2kF (x)TF sin(α(x))



 k (x) cos(2θ (x))
cos(2θF (x))
F
F
QC
−
, for aF < x < xI
n (x) =
12πθF (x)
4kF (x)TF sin(α(x))





e−2|θF (x)|
|kF(x)|e−2|θF (x)|


−
, for x < aF ,

4|kF (x)|TF sinh(|α(x)|)
12π|θF (x)|
(5.53)
(5.54)
where the upper sign means the traveling region, the lower sign the evanescent region, and
αF (x) = πτF (x)/TF .
For tsemi , the subscript a and e mean the traveling and evanescent regions respectively. The
leading term is
kF3 (x)
p
[zF (x)Ai2 (−zF (x)) + Ai′ (−zF (x))2 ]
tsemi
(x)
=
±
ST
2 ±zF (x)
k 3 (x)
∓ F
[2z 2 (x)Ai2 (−zF (x)) − Ai(−zF (x))Ai′ (−zF (x))
9θF (x) F
+2zF (x)Ai′2 (−zF (x))] ,
(5.55)
and the quantum corrections for the KE density are
tQC
(1),a (x)
kF3 (x) cos(2θF (x)) kF (x) cos(2θF (x))
−
=
12πθF (x)
4TF sin(αF (x))
tQC
(2),a (x) = −
p
zF (x)π 2 cos(αF (x))
−
2
5/2
8zF (x) 2kF(x)TF2 sin (αF (x))
kF3 (x)
67
!
(5.56)
Ai2 (−zF (x))
(5.57)
tQC
(1),e (x) =
kF (x)e2θF (x)
kF (x)3 e2θF (x)
−
8TF sinh(αF (x))
24πθF (x)
tQC
(2),e (x) =
(5.58)
!
p
−zF (x)π 2 cosh(αF (x))
kF (x)3
−
Ai2 (−zF (x))
8(−zF (x))5/2
2kF (x)TF2 sinh2 (αF (x))

QC
QC


t(1),a + t(2),a , for xI < x < xmid



1 QC
QC
t (x) =
t(1),a + tQC
(2),a , for aF < x < xI

2




tQC + tQC , for x < aF .
(1),e
(5.59)
(5.60)
(2),e
The total semiclassical density and KE density are
nsemi (x) = nsemi
(x) + nQC (x)
ST
tsemi (x) = tsemi
(x) + tQC (x) .
ST
5.5
(5.61)
1D Applications
5.5.1
Harmonic potential
The Schrödinger equation for a harmonic oscillator is
1
h̄2 d2
2 2
+ mω x ψj (x) = ǫj ψj (x) .
−
2m dx2 2
(5.62)
The exact wavefunction of j-th level is given by
ψj (x) =
r
1 mω 1/4 −mωx2 /2h̄
e
Hj
2j j! πh̄
r
68
mω
x ,
h̄
(5.63)
where Hj is the j-th Hermite polynomial. And the exact energy levels are
1
ǫj = j +
h̄ω ,
2
j=0,1,2,...
(5.64)
In atomic units (m = 1, h̄ = 1), with ω = 1, the wavefunction is reduced to
ψj (x) =
r
1
j
2 j!
1/4
1
2
e−x /2 Hj (x) ,
π
(5.65)
and the exact density for N particles is
n(x) =
N
−1
X
ψj2 (x) .
(5.66)
j=0
EF of the harmonic oscillator for N particles is calculated by
Z
√
2EF
√
− 2EF
p
dx 2(EF − x2 /2) = Nπ ,
(5.67)
√
with EF = N and turning points ± 2N. In Table 5.1, we calculate the percentage errors
√
of the semiclassical density at the turning point (TP, x = − 2N ), using Eqs. (5.53), (5.55)
and (5.61). The leading terms in the density (nsemi
, Eq. (5.53)) and KE density (tsemi
,
ST
ST
Eq. (5.55)) give relatively accurate values at the TP, and the percentage errors decrease as
N becomes large. With quantum corrections, Eq. (5.61) (nsemi and tsemi ) gives very good
results, even for N = 1. The errors are less than 2% and decrease much more rapidly than
those of the leading terms.
We calculate kinetic and potential energies of a harmonic oscillator (v(x) = x2 /2) using
the semiclassical density and KE density in Table 5.2. Errors in the normalization of the
semiclassical densities decrease as N increases. The magnitude of errors for the semiclassical
KE increase with N, but slowly converge to a constant. The errors in potential energies are
69
Table
√ 5.1: The percentage errors of densities and KE densities at the turning point (x =
− 2N) for a harmonic oscillator.
N
1
2
3
4
5
6
7
8
9
10
20
30
40
50
60
70
80
90
100
200
nsemi
ST
24.0717
14.338
10.6836
8.70012
7.43109
6.53896
5.87216
5.35183
4.93263
4.58645
2.85266
2.16591
1.78283
1.53354
1.35623
1.22255
1.11755
1.03249
0.961952
0.604372
nsemi
1.9631
1.50315
1.20183
1.01334
0.884161
0.789477
0.716657
0.658626
0.611107
0.571352
0.365231
0.280327
0.232148
0.200477
0.177797
0.160614
0.147064
0.136056
0.126903
0.0801871
tsemi
ST
13.4611
7.78502
5.71087
4.6052
3.90672
3.42032
3.05945
2.77952
2.55511
2.37056
1.45737
1.10111
0.903817
0.775996
0.685357
0.617179
0.563717
0.520474
0.484651
-
tsemi
1.78836
0.799501
0.482507
0.334419
0.250848
0.198019
0.161993
0.136058
0.116609
0.101555
0.0407332
0.0238158
0.0162631
0.0120943
0.00949334
0.00773511
0.00647718
0.00553832
0.00481425
-
bigger than those in the semiclassical KE and show much slower convergence. In contrast,
the percentage errors in the potential energies go to zero rapidly.
In 2003, Sim et al. derive the kinetic energy density in 1D from the virial theorem for
noninteracting electrons with v-representable densities(77):
DF
t
1
(x) =
2
Z
∞
dx′ n(x′ )
x
dv(x′ )
.
dx′
(5.68)
We test this kinetic energy functional on the semiclassical densities for a harmonic oscillator.
The kinetic energies from Eq. (5.68) have almost the same errors as the potential energies,
satisfying the virial theorem.
To remove the normalization errors, we re-normalize the
semiclassical density to N numerically. With the re-normalized densities, we calculate the
70
Table 5.2: Errors of densities, kinetic energy, potential energy for
(v(x) = x2 /2).
R
R
R
R
N
dx nsemi (x) exact KE
dx tsemi (x)
dx tDF (x)
1
0.9856
0.25
-0.03921639 0.00076356
2
1.9931
1.00
-0.03546662 0.00027550
4
3.9943
4.00
-0.04261346 -0.00195068
8
7.9960
16.00
-0.04766957 -0.00625399
16
15.9974
64.00
-0.05132631 -0.01274630
32
31.9983
256.00
-0.05402198 -0.02092856
64
63.9989
1024.00
-0.05597217 -0.03409107
128
127.9993
4096.00
-0.05753729 -0.04841689
256
255.9996
16384.00 -0.05868514 -0.07050443
a harmonic oscillator
dx nsemi (x)v(x)
0.00075271
0.00027493
-0.00195005
-0.00629082
-0.01273680
-0.02159829
-0.03409480
-0.04943579
-0.07054575
Table 5.3: Errors of kinetic energies and potential energies for a harmonic oscillator (v(x) =
x2 /2). The semiclassical density is re-normalized to N to calculate kinetic energies from
tDF (x) (Eq. (5.68)) and potential energies.
R
R
N exact KE
dx tDF (x)
dx nsemi (x)v(x)
E
1
0.25
0.00447381
0.00441528
0.00888909
2
1.00
0.00376036
0.00375728
0.00751765
4
4.00
0.00374241
0.00373903
0.00748145
8
16.00
0.00192897
0.00173114
0.00366011
16
64.00
-0.00215785
-0.00220972
-0.00436757
32
256.00
-0.00752076
-0.00819052
-0.01571129
64
1024.00 -0.01706026
-0.01706399
-0.03412424
128 4096.00 -0.02716883
-0.02818773
-0.05535656
256 16384.00 -0.04368269
-0.04372401
-0.08740670
kinetic energies using Eq. (5.68) and the potential energies in Table 5.3. The errors in
kinetic and potential energies are similar to those in Table 5.2, but the magnitude of errors
is reduced as N becomes large. We plot the semiclassical (KE) densities with exact results
in Figs. 5.3 and 5.4 for N = 2. Errors present in the semiclassical KE density decrease
rapidly as N increases.
71
Exact
Semi
0.6
n(x)
0.5
0.4
0.3
0.2
0.1
0
-4
-2
0
2
4
x
Figure 5.3: Comparison of exact and semiclassical densities of a harmonic potential with
N = 2.
Exact
Semi
0.5
tS (x)
0.4
0.3
0.2
0.1
0
-4
-2
0
2
4
x
Figure 5.4: Comparison of exact and semiclassical KE densities of a harmonic potential with
N = 2.
5.5.2
Morse potential
Then we apply the semiclassical density and KE density to a Morse potential(78). The
Morse potential is an approximation to the potential energy surface of a diatomic molecule
that gives the better vibrational energy levels than those found using the harmonic oscillator
approximation. An exact solution can be obtained by solving the Schrödinger equation. In
72
this study, we use the following Morse potential:
v(x) = 16(e−x/2 − 2e−x/4 ) .
(5.69)
We obtain the exact density and KE density numerically and compare them with the semiclassical solutions for N = 2 in Figs. 5.5 and 5.6. The semiclassical density in Fig. 5.5
Exact
Semi
0.8
n(x)
0.6
0.4
0.2
0
-2
0
2
4
x
Figure 5.5: Comparison of exact and semiclassical densities of Morse potential with N = 2.
shows very good agreement with the exact density, but the difference of the KE densities in
the traveling region is clear in Fig. 5.6. However, this difference rapidly diminishes as N
increases.
5.6
Is this variational?
According to Ref. (77), we can define the total energy of a given system with the density,
n(x) and the potential, v(x):
1
E=
2
Z
∞
−∞
dx
Z
x
∞
dv(x′ )
+
dx n(x )
dx′
′
′
Z
∞
−∞
73
dx n(x) v(x) .
(5.70)
Exact
Semi
0.8
tS (x)
0.6
0.4
0.2
0
-0.2
-2
0
2
4
x
Figure 5.6: Comparison of exact and semiclassical KE densities of Morse potential with
N = 2.
If we know the density as a functional of a given potential, the total energy will be a functional
of the potential. Hence, we can rewrite the above equation as
1
Ev [v] =
2
Z
∞
dx
−∞
Z
∞
x
dv(x′ )
+
dx n[v](x )
dx′
′
′
Z
∞
dx n[v](x) v(x) .
(5.71)
−∞
Since the kinetic energy functional is derived from the virial theorem, the kinetic and potential energies always satisfy the virial theorem for a given potential and the density from
the potential.
The question then becomes, “Is Eq. (5.71) variational?” To check this, we calculate the total
energies for a given trial potential, ve(x), using
1
Ev [e
v] =
2
Z
∞
−∞
dx
Z
x
∞
de
v(x′ )
dx n[e
v ](x )
+
dx′
′
′
where the trial potential, ve(x), is
1
v (x) = xp , p = 2, 4, 6, and 8 ,
e
2
Z
∞
dx n[e
v ](x) v(x) ,
(5.72)
−∞
(5.73)
74
and the true (external) potential, v(x), is
1
v(x) = x4 .
2
(5.74)
In Table 5.4, we summarize the KE, PE, and total energies of N = 1, 2, 3, and 4 for given
trial potentials. We re-normalize the semiclassical densities numerically to N. The minimum
Table 5.4: Variational principle. The trial potential, ve(x), is xp /2 and the exact (target)
potential is x4 /2. For all cases, the minimum of the total energy is at p = 4.
N =1
N =2
N =3
N =4
p
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
KE
0.25447381
0.36936501
0.47279080
0.56578485
1.00376036
1.63598630
2.11907692
2.54624093
2.25548094
4.12831726
5.53804028
6.69177584
4.00374241
8.00986449
11.14843359
13.66074179
PE
0.38465361
0.18467997
0.13254050
0.10765556
2.26951144
0.81775834
0.52224034
0.40335159
7.16249997
2.06415038
1.18722743
0.86496638
16.54509916
4.00454536
2.11463851
1.46596067
E
0.63912742
0.55404498
0.60533129
0.67344041
3.27327181
2.45374464
2.64131726
2.94959252
9.41798091
6.19246764
6.72526771
7.55674222
20.54884157
12.01440985
13.26307210
15.12670246
KE/PE
2.00003
2.00057
2.00001
2.00019
of the total energy for all N is always at p = 4.
We also test the variational principle by changing the coefficient of x2 in Table 5.5. The trial
and true potentials are, respectively,
v (x) = αx2 ,
e
v(x) =
3 2
x .
4
(5.75)
75
We change the α from 1/2 to 1.0 and calculate the total energies using Eq. (5.72) for trial
potentials. The energy minimum is found at α = 3/4.
Table 5.5: Variational principle for N = 1. The trial potential, ve(x), is αx2 and the exact
(target) potential is 3x2 /4. The minimum of the total energy is at α = 3/4.
α
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
5.7
KE
0.35979927
0.35068325
0.34148517
0.33183181
0.32187217
0.31159691
0.30225659
0.29017028
0.27872824
0.26685058
0.25447380
PE
0.26984201
0.27685260
0.28444855
0.29267613
0.30169775
0.31158810
0.32262455
0.33470732
0.34836102
0.36385248
0.38162292
E
0.62964128
0.62753585
0.62593372
0.62450794
0.62356993
0.62318501
0.62488114
0.62487760
0.62708926
0.63070306
0.63609672
KE/PE
1.00003
Spherically symmetric 3D potential
In this section, we discuss the non-interacting, spinless fermions in spherically symmetric
potentials in 3D. The Hamiltonian for the system is given in atomic units as
1
Ĥ = − ∇2 + v(r) .
2
(5.76)
The Laplacian in spherical coordinates is expressed as
∇2 =
∂2
2 ∂
1
+
− 2 L̂2 .
2
∂r
r ∂r r
(5.77)
76
We can separate the wavefunction ψ(r) into a radial, R(r), and an angular part, Ylm (θ, φ):
ψ(r) = Rnl (r)Ylm (θ, φ) ,
(5.78)
where Ylm (θ, φ) is the eigenfunction of L̂2 :
L̂2 Ylm (θ, φ) = l(l + 1)Ylm (θ, φ) .
(5.79)
The radial part of Schrödinger equation becomes
1
−
2
2 ′
l(l + 1)
′′
Rnl (r) + Rnl (r) +
+ v(r) Rnl (r) = ERnl (r) ,
r
2r 2
(5.80)
where this is the eigenvalue problem in 1D. To remove the singularity at the origin, we can
define the radial wavefunction as
Rnl (r) =
unl (r)
.
r
(5.81)
Then, the differential equation can be further simplified to yield
l(l + 1)
d2
unl (r) = 0 .
+ 2 E − v(r) −
dr 2
2r 2
(5.82)
For a given spherical potential v(r), the exact radial density is the sum of u2nl (r):
2
4πr n(r) =
n X
i−1
X
u2ij (r) .
(5.83)
i=1 j=0
The exact centrifugal potential, l(l + 1)/2r 2 , does not give the right orbital energy or asymptotic behavior of the semiclassical wavefunctions for the Coulomb potential, −Z/r(75; 79).
Thus, Langer(75) proposed the following modification of the angular contribution, which en-
77
sures the right energy and the asymptotic decays of semiclassical wavefunctions(79; 80; 81)
2
1
l(l + 1) → l +
.
2
(5.84)
For l = 0, this modification converts the single-turning-point problem into the two-turningpoint problem. According to this modification, the effective potential can be rewritten as
v eff (r) = v(r) +
(l + 1/2)2
,
2r 2
(5.85)
and Eq. (5.82) can be written as
d2
eff
+ 2[E − v (r)] unl (r) = 0 ,
dr 2
(5.86)
which is a 1D Schrödinger equation with two turning points. Hence, we can apply our
semiclassical density formula to Eq. (5.86). Similar to the separation of space shown in Fig.
5.2, we define the necessary points for the semiclassical scheme in Fig. 5.7. r+ and r− are
Potential
EF
Radial density
4πr2 n(r)
0.2
0.1
0
-0.1
r+
-0.2
0
rmid
rI
2
4
6
r−
rII
8
10
12
14
16
r
Figure 5.7: Spherical 3D systems.
78
the turning points at EF and the points, rI and rII, are found by the conditions
θF (rI ) =
θF (rII) =
Z
rI
r
Z +r−
dr kl (r; EF ) = π/4 ,
dr kl (r; EF) = π/4 .
(5.87)
rII
Our semiclassical equations give the radial density. Therefore, 4πr 2 appears in front of
QC
nsemi
(r) to prevent confusion between these and the previous 1D results.
ST (r) and n
5.7.1
Isotropic 3D harmonic potential
The Schrödinger equation for an isotropic 3D harmonic oscillator is given by
1
1
− ∇2 ψ(r) + r 2 ψ(r) = Eψ(r) .
2
2
(5.88)
The exact radial wavefunction is
Rnl (r) = Nnl r l e−r
2 /2
l+1/2
L(n−l)/2 (r 2 ) ,
(5.89)
l+1/2
where L(n−l)/2 (r 2 ) is an associated Laguerre polynomial and Nnl is the normalization constant
Nnl =
s
n+l ! 2 !
2n+l+2 n−l
2
√
.
π(n + l + 1)!
(5.90)
The energy levels are given by En = (n + 3/2), which is (n + 1)(n + 2)/2-fold degenerate.
With angular momentum quantum number l, the possible principle quantum number n is l,
l + 2, l + 4 . . . . We use Langer’s modification for the centrifugal potential. So the effective
79
potential is given by
1
(l + 1/2)2
v eff (r) = r 2 +
.
2
2r 2
(5.91)
Near the origin, 4πr 2nQC (r) (Eq. (5.54)) overcorrects 4πr 2 nsemi
(r) (Eq. (5.53)), hence
ST
4πr 2 nsemi (r) can be negative for radial densities. This becomes severe in s and p orbitals,
compared to d and f orbitals, which is due to the rapid change of the centrifugal potential,
1/r 2 , near the origin. To avoid this, we use the single-turning-point formula near the origin:
4πr 2 nsemi
app (r) =
2zF (r)
[zF (r)Ai2 (−zF (r)) + Ai′ (−zF (r))2 ] .
kF (r)τF (r)
(5.92)
Kohn and Sham(74) suggest the linear equation around the turning point:
4πr 2 nlin(r) = |2v ′ (aF )|1/3 (zF (r)Ai2 (−zF (r)) + Ai′2 (−zF (r))) .
(5.93)
If we take Taylor series of the potential at aF , Eq. (5.93) can be derived from Eq. (5.92). However, Eq. (5.92) is not continuous with the original semiclassical approximation 4πr 2nsemi (r)
2 semi
(Eq. 5.61). We calculate the ratio between 4πr 2 nsemi
(rI )
app (rI ) of Eq. (5.92) and 4πr n
of Eq. (5.61), and multiply the radial density of Eq. (5.92) by the ratio at r = rI . For
r > rI , we use our semiclassical approximation (Eq. (5.61)) to produce the right quantum
oscillations. We apply this method to the 3D spherical harmonic oscillator in Figs. 5.8, 5.9,
and 5.10, demonstrating that our semiclassical approximation produces very accurate radial
densities.
80
1.5
4πr2 n(r)
Exact
Semi
1
0.5
0
0
1
2
3
4
5
r
Figure 5.8: Comparison of exact and semiclassical densities of l = 0 orbitals with N = 3 for
an isotropic 3D harmonic potential. Eq. (5.92) is used from r = 0 to rI .
1.5
4πr2 n(r)
Exact
Semi
1
0.5
0
0
1
2
3
4
5
r
Figure 5.9: Comparison of exact and semiclassical densities of l = 1 orbitals with N = 3 for
an isotropic 3D harmonic potential. Eq. (5.92) is used from r = 0 to rI .
5.7.2
Bohr atom
The Hamiltonian for a hydrogenic system is given by in atomic units
1
Z
Ĥ = − ∇2 − ,
2
r
(5.94)
81
1.5
4πr2 n(r)
Exact
Semi
1
0.5
0
0
1
2
3
4
5
r
Figure 5.10: Comparison of exact and semiclassical densities of l = 2 orbitals with N = 3
for an isotropic 3D harmonic potential. Eq. (5.92) is used from r = 0 to rI .
where Z is the atomic number. We can separate the wavefunction ψ(r) into a radial, R(r),
and an angular part, Ylm (θ, φ):
ψ(r) = R(r)Ylm (θ, φ) .
(5.95)
The radial part of Schrödinger equation becomes
1
−
2
2 ′
l(l + 1) Z
′′
R (r) + R (r) +
R(r) = ER(r) .
−
r
2r 2
r
(5.96)
The exact radial solution for the above equation is given as
R(r) = Nnl
2Zr
n
l
−Zr/n
e
L2l+1
n−l−1
2Zr
n
,
(5.97)
where n is the principal quantum number, L2l+1
n−l−1 is the associated Laguerre polynomial and
Nnl is a normalization constant,
Nnl =
s
(n − l − 1)!
2n(n + 1)!
2Z
n
3
.
(5.98)
82
We define the radial wavefunction as
R(r) =
u(r)
.
r
(5.99)
Then, the differential equation can be simplified to
Z l(l + 1)
d2
+2 E+ −
u(r) = 0 .
dr 2
r
2r 2
(5.100)
According to Langer’s modification(75), the effective Coulomb potential can be rewritten as
v eff (r) = −
Z (l + 1/2)2
+
,
r
2r 2
(5.101)
and the semiclassical quantities are defined for a given angular momentum quantum number
l as
kl (r) =
θl (r) =
τl (r) =
p
Z
2(E − v eff (r))
r
r
Z +r
r+
dr ′ kl (r ′ )
dr ′
1
.
kl (r ′ )
(5.102)
The EF is determined by the following quantization condition:
Z
r−
r+
dr kl (r; EF ) = (n − l)π ,
(5.103)
where n and l are the valence principal and angular momentum quantum numbers respec-
83
tively, and the turning points at the EF are given by
r−
p
Z 2 + 2EF (l + 1/2)2
2EF
p
2
−Z − Z + 2EF (l + 1/2)2
.
=
2EF
r+ =
−Z +
(5.104)
Unlike the isotropic 3D harmonic oscillator, the potential, v eff (r), goes to zero as r → ∞,
and the EF for given n and l is generally close to zero. In this case, the leading term,
2 QC
4πr 2 nsemi
(r), Eq. (5.54), after r− ,
ST (r) (Eq. (5.53)), decays too rapidly compared to 4πr n
and the semiclassical radial density becomes negative in the right-side evanescent region.
2 semi
However, 4πr 2 nsemi
app (r) (Eq. (5.92)) decays slowly compared to 4πr nST (r). In addition,
2 semi
4πr 2 nsemi
(r− ) = 4πr 2 nsemi
ST
app (r− ) at the turning point. Hence, we use 4πr napp (r) without any
quantum corrections in the regions, r < rI and with quantum corrections (4πr 2 nQC (r)) in
the region for which r− < r.
Exact
Semi
4πr2 n(r)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
r
Figure 5.11: Comparison of exact and semiclassical densities of 1s1 2s1 3s1 4s1 .
We plot the exact and semiclassical radial densities with Z = 1 in Figs. 5.11, 5.12, 5.13
and 5.14. For example, if the valence quantum number n = 4 and l = 0, this means 4
84
Exact
Semi
4πr2 n(r)
0.25
0.2
0.15
0.1
0.05
0
0
10
20
30
40
50
r
Figure 5.12: Comparison of exact and semiclassical densities of 2p1 3p1 4p1 .
(=n − l) particles, and its electronic configuration for non-interacting same spin electrons
is 1s1 2s1 3s1 4s1 . The semiclassical formalism for the given potential, v eff (r), gives the right
0.2
4πr2 n(r)
Exact
Semi
0.15
0.1
0.05
0
0
10
20
30
40
50
r
Figure 5.13: Comparison of exact and semiclassical densities of 3d1 4d1 .
quantum oscillations for p, d, and f orbital densities, compared to the exact radial density.
However, in Fig. 5.11, a few oscillations near the origin show significant differences in the
s-orbital density. Also, there is a slight phase shift for the s-orbital density which can also
be seen in Fig. 9 of Ref. (74), where WKB wavefunctions were used. This is due to Langer’s
transformation and will be discussed in Section 5.8.
85
0.07
Exact
Semi
4πr2 n(r)
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
r
Figure 5.14: Comparison of exact and semiclassical densities of 4f 1.
5.8
Real atoms
5.8.1
Single-point calculation
We begin with the KS formalism. The KS equations for any atoms or ions are
1 2
− ∇ + vS σ (r) φiσ (r) = ǫiσ φiσ (r) ,
2
(5.105)
and the spin densities are
nσ (r) =
Nσ
X
i=1
|φiσ (r)|2 ,
(5.106)
where vS σ (r) is a single, multiplicative spin-dependent KS potential, and σ is the spin index(↑
and ↓). The spin-dependent KS potential is a sum of three contributions:
vS σ (r) = v(r) + vH [n](r) + vXC σ [n↑ , n↓ ](r) ,
86
(5.107)
where v(r) = −Z/r for an atomic system, vH (r) is the Hartree potential given as
vH (r) =
Z
d3 r ′
n(r′ )
,
|r − r′ |
(5.108)
and the exchange-correlation (XC) contribution is
vXC σ [n↑ , n↓ ](r) =
δEXC [n↑ , n↓ ]
.
δnσ (r)
(5.109)
For an atomic system, the KS equations can be separated into the radial and angular parts:
1 2
2 ∂
1 ∂2
+ 2 L̂ + vS σ (r) φiσ (r) = ǫiσ φiσ (r) .
+
−
2 ∂r 2 r ∂r
2r
(5.110)
eff
This can be further reduced to a 1D radial equation like Eq. (5.82). Let us define vsσ
(r)
with this centrifugal potential:
eff
vsσ
(r; l) = vext (r) + vH [n](r) + vXC σ [n↑ , n↓ ](r) +
l(l + 1)
.
2r 2
(5.111)
This exact centrifugal potential does not produce the right asymptotic behavior of semiclassical wavefunctions discussed in Section 5.7. Thus we use Langer’s modification instead of
the exact l(l + 1)
eff
vsσ
(r; l) = vext (r) + vH [n](r) + vXC σ [n↑ , n↓ ](r) +
(l + 1/2)2
,
2r 2
(5.112)
eff
where vsσ
(r) always has two turning points even for l = 0.
We obtain the total radial density (4πr 2 n(r)) and the KS potential (vS (r)) of a neutral Kr
atom (Z = 36) with the LDA XC functional using the fully numerical OPMKS code(38).
The electronic configuration of Kr is [Ar]3d10 4s2 4p6 . Since Kr is a spin-unpolarized, fully
occupied atom, the KS potentials for up-spins and down-spins are identical. With vSLDA (r)
87
and Eq. (5.112), we calculate EF for a given l:
Z
r−
r+
q
eff
2(EF − vs↑
(r; l)dr = (n − l)π ,
(5.113)
where {(n, l)} = {(4, 0), (4, 1), (3, 2)}. For example, for the up-spin s-orbital density, n = 4
and l = 0. This means that the number of up-spin electrons up to 4s is 4(= n − l). With this
EF , we apply our semiclassical scheme to calculate the radial density (4πr 2nsemi
n,l (r)) for given
n and l. Degeneracy by magnetic quantum numbers (m) and spins (mS ) are handled by
multiplication of the semiclassical radial density by 2(2l + 1). Hence, the total semiclassical
radial density will be the sum of each of the radial densities for the given angular momentum
quantum numbers:
4πr 2 nsemi (r) =
X
2(2l + 1)4πr 2nsemi
n,l (r) .
(5.114)
{n,l}
We compare the semiclassical result with SCF LDA density from OPMKS code in Fig. 5.15
60
SCF-LDA
Semi
4πr2 n(r)
50
40
30
20
10
0
0
1
2
3
4
r
Figure 5.15: Comparison of SCF-LDA and semiclassical densities of Kr. Eq. (5.93) is used
from r = 0 to rI . (l + 1/2)2 is used for all orbitals instead of l(l + 1).
and find they show good agreement, superficially. However, if we enlarge Fig. 5.15 and focus
on the region where r < 1.2, we can find the phase shift and incorrect amplitude visible in
88
Fig. 5.16. Langer(75) stated that, “Whenever the number l is small the turning point lies
4πr2 n(r)
50
SCF-LDA
Semi
40
30
20
10
0.2
0.4
0.6
0.8
1
1.2
r
Figure 5.16: Comparison of SCF-LDA and semiclassical densities of Kr. Eq. (5.93) is used
from r = 0 to rI . (l + 1/2)2 is used for all orbitals instead of l(l + 1).
very near to the point r = 0, and this is true also in the case of the turning point whenever
the energy is very large. The use of the formulas under such circumstances is questionable,
and is generally less satisfactory than otherwise.” Also, Berry and Mount (79) mentioned
“inaccuracies in Langer’s method for low l, when the turning point after the modification
comes very close to the origin.”
We here adjust Langer’s modification for s and p orbitals to eliminate these effects. We use
(l + 0.53)2 for l = 0 and (l + 0.51)2 for l = 1. Although we do not present the result in this
dissertation, the semiclassical radial density becomes more accurate than the density with
Langer’s original modification. This refinement seems to work for other atoms, but we need
to show that this adjustment is justified.
5.8.2
Semiclassical orbital-free potential-density functional
Since vS σ (r) in Eq. (5.110) depends on the (spin)-density, we solve the KS equations iteratively and find a self-consistent solution. The flow chart of the KS-SCF procedure is depicted
89
in Fig. 5.17. By applying the Hohenberg-Kohn(2) theorem to the non-interacting system,
Figure 5.17: SCF procedures for KS-DFT (left) and orbital-free potential-density functional
theory (right).
the KS potential, vS σ (r), which gives the ground-state spin density, nσ (r) for the system, is
unique. If nσ (r) can be expressed as a functional of vS σ (r), we can state
1:1
nσ [vS σ ](r) ←→ vS σ [nσ ](r) .
(5.115)
Using our semiclassical approximation to the density as a potential functional, we can directly
obtain the radial density from a given KS potential without ever solving the KS equations.
Since vS σ (r) is a density functional, we have to find a self-consistent solution. Hence we set
up a new SCF procedure for orbital-free potential-density functional theory (OF-pDFT) in
Fig. 5.17.
For OF-pDFT calculations, we need an initial guess for the density in order to construct
vS σ (r). The initial density comes from a SCF calculation with an exact exchange (exchange-
90
only) (18) functional using OPMKS code(38) for Ar. In our procedure, the potential for a
given l is defined as
eff
vsσ
(r; l) = vext (r) + vH [n](r) + vXC σ [n↑ , n↓ ](r) +
(l + 1/2)2
,
2r 2
(5.116)
where vXC (r) is the LDA exchange(29; 30) and correlation (VWN5)(31) functional, vext is
−Z/r, and vH is the Hartree potential. For closed-shell atoms, we use the spin-unpolarized
LDA functional, and the Hartree potential can be obtained without solving the Poisson
equation:
Z 1
vH (r) = −
r
r
Z
∞
r
dr
′
Z
∞
dr ′′ 4πr ′′n(r ′′ ) .
(5.117)
r′
So, from the initial density, n0 (r), we calculate the LDA KS potential. Ar has [Ne]3s2 3p6 as
its electronic configuration. To calculate the semiclassical radial density, we need {n, l} =
{(3, 0), (3, 1)}. Then the total semiclassical radial density with the LDA KS potential will
be
4πr 2 nsemi (r) =
X
2(2l + 1)4πr 2nsemi
n,l (r) .
(5.118)
{n,l}
The convergence criteria is the density difference (i.e., ∆n(r) < 10−5 ) between the previous
and current iterations. If this condition is satisfied, then SCF is terminated and, if not, the
SCF procedure will be repeated to achieve the convergence criteria.
In Fig. 5.18, we plot the LDA-SCF radial density and the converged semiclassical radial
density from OF-pDFT. The semiclassical radial density still shows a slight phase shift, but
this is due to the inaccuracy in Langer’s method discussed in the previous Section 5.8.1.
We can also define the total energy as a potential functional. In Ref. (77), the kinetic energy
for 3D spherical systems can be obtained from the virial theorem for the v-representable non91
25
4πr2 n(r)
20
15
10
5
0
0
1
2
3
4
5
r
Figure 5.18: Comparison of LDA-SCF and semiclassical SCF densities of Ar.
-565.8
E
-566
-566.2
-566.4
0
10
20
30
40
50
N
Figure 5.19: Convergence of total energies in each iterations
interacting density:
TsDF [vS ]
3
=
2
Z
3
dr
Z
r
∞
dr ′n[vS ](r ′ )
dvs (r ′ )
.
dr ′
(5.119)
So the total energy of the spherical system is
E[vS ] = TsDF [vS ] + Eext [n[vS ]] + EH [n[vS ]] + EXC [n[vS ]] .
(5.120)
We plot the total energy from each iteration in Fig. 5.19. Since we use the approximate XC
92
(LDA) functional, the total energies are not variational. However, the total energies almost
converge after 30 iterations and the energy difference is less than 4.8 × 10−6 H.
93
Chapter 6
Conclusion
In this dissertation, I have discussed three topics. I first explained how reasonably accurate
electron affinities can be obtained via approximate density functionals. Incomplete cancellation of self-interaction in approximate density functionals results in positive HOMO energies
and thus, unbound HOMO state. By studying the KS and XC potentials, I show that shifting
the KS potential by a constant does not affect the total density and, therefore, the meaning
of the positive HOMO becomes ambiguous. Hence, I show that an accurate density and
EA can be obtained with density functional approximations, despite the positive HOMO
energy. In addition, I suggest a simple and practical way to evaluate the EAs on HF and
EXX densities that bind a valence electron. The results from HF and EXX densities are
reasonable and slightly better than those of the traditional approaches. However, if the HF
density inaccurately describes the system due to a lack of correlation, then our method will
fail to get accurate EAs.
Second, I gave a condition on the KS kinetic energy. It is shown here that the asymptotic
expansion of total energies with atomic number Z gives a vital condition that the noninteracting KS kinetic energy should satisfy. I construct the modified gradient expansion
94
approximation to the fourth order, which gives the correct asymptotic expansion coefficients.
I apply this modified GEA to atoms, molecules, and the jellium surface/sphere. I find that,
for atoms and molecules, our modified GEA improves the kinetic energies, when compared
to those generated with TF and original GEA kinetic energy functionals. However, for the
atomization process, the TF kinetic energy functional gives better mean absolute errors
(0.25) than GEAs. This is because the GEA does not include quantum corrections by the
turning points. I have also studied the existing kinetic energy functional approximations
and found the corresponding asymptotic expansion coefficients. This modified GEA cannot
be used in orbital-free density functional calculation, since if I take the functional derivative
of the fourth order modified GEA to evaluate the kinetic energy potential, the derivatives of
the fourth and higher terms will diverge for atoms. Furthermore, if I include the sixth order
term in the GEA to improve the accuracy, the sixth order term of T GEA will also diverge
for atoms. Hence, the development of kinetic energy functionals based on the GEA is not
appropriate.
In the final section, the uniform semiclassical Green’s function using Langer’s uniformization
is constructed. Using a specifically chosen contour, I have derived the semiclassical density
and kinetic energy density as potential functionals for 1D systems with turning points. I have
demonstrated the accuracy of our semiclassical formalism for both the harmonic oscillator
and the Morse potential. With the kinetic energy functional from the virial theorem, I can
define the total energy as a potential functional. Since spherical 3D systems such as 3D
spherical harmonic oscillators and Bohr atoms can be reduced to quasi-1D systems, I can
apply our semiclassical scheme to obtain radial densities.
Also the radial part of the KS equation for real atoms is a 1D differential equation. The KS
potential is a functional of the density, and the semiclassical radial density is a functional
of the KS potential. Hence, I can set up a SCF procedure to find the density and the
potential. For Ar, I have used the LDA XC functional for the KS potential and found the
95
converged, self-consistent, semiclassical radial density. The kinetic energy functional of a
spherical system can also be obtained from the virial theorem, so I define the total energy
potential functional for real, spherical atoms and get the converged total energy using this
functional.
The semiclassical scheme is only applicable to 1D and spherical 3D systems. For general
3D problems including non-spherical systems, I need to explore other approaches, such as a
path integral formalism and Gutzwillers semiclassical Green’s function(82). There are many
accurate approximations to EXC [n] such as GGAs for general 3D systems. This means that
a highly accurate approximation to the density as a potential functional automatically leads
to an electronic structure method that scales linearly with the number of particles, N. I will
then be able to study large biological molecules with orbital-free potential-density functional
theory, avoiding any QM/MM or empirical force field methods.
96
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100
Appendices
A
Construction of modified GEA
To construct the modified GEA (MGEA) which gives the second and third coefficients in the
asymptotic expansion exactly, we multiply the T (2) and T (4) terms of the gradient expansion
by new coefficients a and b respectively, i.e. a = 1, b = 0 would simply be the second order
GEA.
T M GEA = T (0) + aT (2) + bT (4)
(A.1)
T M GEA − T (0) = aT (2) + bT (4)
(A.2)
5
GEA
2
M GEA
3
= (cM
− cTF
− cTF
1
1 )Z + (c2
2 )Z
(A.3)
GEA
If a=1 and b=0, then the above equation gives T2nd
GEA
T (2) = T2nd
− T (0)
5
2
GEA2
3
= (cGEA2
− cTF
− cTF
1
1 )Z + (c2
2 )Z
101
(A.4)
GEA
If a=1 and b=1, then the above equation gives T4th
GEA
T (2) + T (4) = T4th
− T (0)
5
2
GEA4
3
= (cGEA4
− cTF
− cTF
1
1 )Z + (c2
2 )Z
(A.5)
According to the linearity between the coefficients of Z-expansion and the 2nd/4th GEA,
TF
then there will be a matrix to map (a,b) into (c1 -cTF
1 , c2 -c2 )





TF
 c1 − c1   m11 m12   a 
 
=

TF
b
m21 m22
c2 − c2



cGEA2
1
cGEA2
2
−
cTF
1
−
cTF
2

(A.6)




(A.7)




(A.8)
  m11 m12   1 
 
=
0
m21 m22
GEA4
− cTF
1
  m11 m12   1 
 c1
 
=

GEA4
TF
1
m21 m22
c2
− c2
Hence,



cGEA2
1
cTF
1
cGEA4
1
cGEA2
1

−
−

 m11 m12  

=

TF
GEA4
GEA2
cGEA2
−
c
c
−
c
m21 m22
2
2
2
2
(A.9)
Using the coefficients in Table. 4.1, I could get the M matrix and the corresponding (a,b).




 m11 m12   0.1246 0.0162 

=

−0.0494 0.0071
m21 m22
102
(A.10)
GEA M GEA
GEA M GEA
If (cM
,c2
) are chosen to give exact expansion, i.e. (cM
,c2
)=(-0.5, 0.2699),
1
1
then




GEA
cM
1
−
cTF
1
GEA
cM
− cTF
2
2





  0.1608 

=
−0.1155
(A.11)

 a   1.789 
 =

b
−3.841
(A.12)
Using these coefficients, we can find:
T M GEA4[n] = T TF [n] + 1.789 T (2) [n] − 3.841 T (4) [n] .
103
(A.13)